Paris: Place de la Concorde

The Place de la Concorde is one of the major public squares in Paris, France. And was through the french revolution 1 of the bloodiest places, thanks to the many executions with the guillotine. Spectators of these executions where literately standing in blood.

Paris: Hôtel de Ville

Since at least the 13th century, the Place was a site where Parisians gathered not to enjoy music or other festivities, but to witness their countrymen and women suffer slow, agonizing deaths.

Cambodia: Khmer Rouge Killing Fields

Analysis of 20,000 mass grave sites by the DC-Cam Mapping Program and Yale University indicate at least 1,386,734 victims of execution.[1][2] Estimates of the total number of deaths resulting from Khmer Rouge policies, including disease and starvation, range from 1.7 to 2.5 million out of a 1975 population of roughly 8 million. In 1979, Vietnam invaded Democratic Kampuchea and toppled the Khmer Rouge regime.

Congo: Congo Free State (1885 to 1908)

In the period from 1885 to 1908, a number of well-documented atrocities were perpetrated in the Congo Free State (today the Democratic Republic of the Congo) which, at the time, was a colony under the personal rule of King Leopold II of Belgium. These atrocities were sometimes collectively referred to by European contemporaries as the "Congo Horrors", and were particularly associated with the labour policies used to collect natural rubber for export. Together with epidemic disease, famine, and a falling birth rate caused by these disruptions, the atrocities contributed to a sharp decline in the Congolese population. The magnitude of the population fall over the period is disputed, but it is thought to be between one and 15 million people.

If one accepts its historicity, the Kurukshetra War, as described in the Mahabharata, which unfolded in the small fields of Kurukshetra.

According to some sources, almost all of the 7 Akshauhinis (1,530,900 soldiers) on Pandava's side and the 11 Akshauhinis (2,405,700 soldiers) on the Kaurava's side perished.

According to another source, the Stri Parva (Mbh 9.1, Mbh 11.26), Yudhishtira provides the following report on casualties and survivors to Dhritarashtra:

1 billion, 660 million, and 20 thousand men perished in the war. Only 240,165 heroes survived.

It is impossible to answer this question beyond personal opinion, so it is a question best tackled by describing the considerations for answering it first.

The substantial problem is that the bulk of history from the birth of Homo sapiens to now is lost – if not completely (pre-history), then for the detail of the events and the veracity of the accounts (pre-modern history). Consequently, (and in step with the research of Keeley, War Before Civilization, and Pinker, The Better Angels of our Nature), it is natural to want to assign too much weight to pre- and early history. This approach, great as it may be for answering questions on trends of violence and the murderous intent of human society, is out of step with the bit we know of the history of human population expansions and densities. For instance, at one time Constantinople was one of the largest concentrations of humans on the planet, and at the time, its population was well below the one-million mark – it had less people than modern Birmingham.

From this observation, the question is not of populations, but population densities per square mile in relation to a violence factor: being not only the number of wars, but the actual duration of the waging of these wars per each of the square miles in a preferred time unit – probably day units, rather than year units. For example, Stalingrad was completely razed during the Second World War, but it did not see violence for the entire duration of the war. As an additional dimension, the traditional invasion route from Europe into Russia had witnessed many wars, but the blood effect of the Napoleonic campaign of 1812 must have been considerably less than the German invasion and occupation. This is entirely due to the industrialisation of warfare. By the same token, the efficiency in killing is not purely attributable to means. It is more appropriate to estimate these means in relation to the delta between that of the one side vis à vis the opposing side, coupled with the desire to kill. For instance, the Asiatic nomadic means of warfare was extremely efficient for its time, and an untold number of deaths followed as the result.

Where is the Terrible Square Mile? Based on the preceding, it is possible to construct an abstract function for the calculation thereof, even as only one of the primitives is quantifiable: f(D, M, W), where D = population density for each square mile of the habitable surface since the dawn of our species, M = efficiency of killing means; and W = the strength of the will. However, as stated, in the absence of a perfect knowledge of history regarding D, our inability to measure two of the primitives of this function is the least of our issues.

Therefore, and purely as a guess, I’d say it is probably located somewhere in a lozenge shaped area starting at Smolensk and reaching towards Moscow.

Tokyo

Simply because Tokyo hast the highest population density of any industrial city with more than 100,000 inhabitants per square mile during the Second World War, the air raids on Tokyo might have placed the bloodiest square mile caused by conventional warfare in this area.

The firebombing raid in the night of 9/10 March 1945 alone caused approximately 100,000 deaths in central Tokyo (an area of about 16 square miles).

Jerusalem

A source says that the so-called “City of Peace” has been

destroyed at least twice, besieged 23 times, attacked an additional 52 times, and captured and recaptured 44 times.

Some square mile of The Somme, probably around some WWI strongpoint.

While it's impossible to determine how many casualties happened at any particular square mile of the Somme, the concentration of so many military casualties in such a small area is something the world had never seen, and never saw again.

Note: "causalities" includes all "rendered unfit for duty" and dead.

The first Battle of the Somme in 1916 had some 1.1 million military casualties concentrated along a 20 mile line.

Then the northern half of Operation Michael attacked at the Somme again to the tune of another 500,000 casualties for the whole offensive.

Finally, August 1918 saw Allied armies counter-attack at the Second Battle of the Somme with relatively light casualties, "only" about 30,000.

That's some 1.6 million casualties over a line of perhaps 20 miles.

The rigid and linear nature of WWI fighting (ie. soldiers in two trenches attacking across a narrow front), scale of soldiers involved, deadliness of weapons, and inadequacy of tactics lead to outrageous casualties along a 20 mile line.

While there are battles which have reached that scale of casualties, the Battle of Moscow at about 1.2 million for example, or the Siege of Leningrad at anywhere from 1.1 million to 4.5 million (many from non-combat reasons), none were all military casualties concentrated in such a small area. And certainly there are campaigns which have had far more casualties, these were spread over a wide area.

And while there have been patches of land that have seen death for much longer periods, the concentrations of casualties dwarf anything before the Somme. There wasn't the population to kill, and there wasn't the infrastructure to concentrate them in a square mile and keep them from running away.

We can't be sure

The much lower population and population density of antiquity means that the least documented parts of history probably aren't in the running. There have been (Fermi estimate) about 100 billion humans all told. Pace the nonsense taught to mid-20th century anthropologists, hunter–gatherer societies live in states of constant war with more than a ¼ of the men eventually dying by homicide. You could imagine an African watering hole, the Nile Valley, or an oasis in the Sinai as earning pride of place by millennia upon millennia of raids and assaults. However, out of the roughly 100 billion humans to be born on Earth, only 1 billion is thought to have preceded the onset of agriculture and the vast majority of them would have died as infants. You're probably looking at less than 60 million homicides spread out over the entire surface of Afroeurasia.

On the other hand, the prejudice towards 20th century death tolls in most of these answers probably isn't completely warranted, either. Once agriculture started, numbers surged. More than 40% of humanity had come and gone before the year 1; more than 60% by the year 1200; ~85% before 1900. All of this while homicide rates were far higher. Almost any large city continuously inhabited through all of this could rack up large numbers just from crime, endurance, and the fairly high density within walled compounds.

Since no one has mentioned them, I'll just toss some Chinese cities into the mix, although the present-day Sinosphere seems to have lower than average homicide rates with the PRC well below where its per capita income would usually place things and industrialized Japan as the lowest in the world.

Xi'an

Has moved around a bit but has been settled since the Neolithic. It's usually been among the world's largest cities (100k–1m) for the past three thousand years and seen plenty of violence beyond the standard, as possibly the most sacked city in the world (58+ times). Those sacks repeatedly involved massacres, as with Huang Chao's in the later Tang that killed so many people that when the city was refounded afterwards it covered 1/16 its former area. It was relatively sedate in the 20th century, though, apart from the genocide of ~20k Manchus during the Xinhai Revolution.

Nanjing

Nanjing is a relative newcomer in China, with Yuecheng having been settled about 2500 years ago but served as a regional or national capital since then. It fell less often and was less inhabited than Xi'an through most of its history, but it's suffered more more recently. The Taiping capture of the city involved about 40k deaths, with another 40k during each of the two failed Qing assaults, followed by around 200k combatant deaths and the slaughter of most of the city during the successful Qing recapture in 1864. The size of the Nanjing Massacre in WWII is a political football but involved the death of most of the city and was likely multiple hundreds of thousands.

Yangzhou

Nanjing had been spared during the Qing conquest of China because it promptly surrendered. It did so after the Qing had utterly butchered the population of the nearby metropolis of Yangzhou, traditionally held to have been 800k. (300k is a modern estimate.) A separate massacre had butchered the city's foreign merchants during the An Shi Rebellion against the Tang.

Guangzhou

was in place by about 3000 years ago and remained the regional capital after its conquest by Shi Huangdi's general Zhao Tuo 800 years later. Sacked by Muslims in 758, followed by Huang Chao's massacre of 100–200k in the foreign ghettos during the later Tang.

Strong contenders would likely include the deadlier major WW1 battlefields (Passchendaele, Verdun, Somme, etc.) - they were more spread out than a single square mile, but the amount of casualties in them was record setting.

Auschwitz-Birkenau (~1.1M deaths) and Treblinka (~700-900k deaths) are also worth a reference, with the latter "beating" the former in terms of deaths per month. These aren't technically deaths by conventional weapons (not all of them anyway) but I feel no answer would be complete without mentioning how the nazis industrialized death.

Edit: Seeing your comment about gas chambers not counting and this being for a story, another genocide you might want to dig into is that of Rwanda, a tiny country (smaller than the New York metropolitan area) in Central Africa. It was an utter bloodbath that led to an estimated 500k-1M deaths using machetes, clubs, blunt objects, and other weapons over a 100-day period. (For perspective, that rate of death is faster than that seen in Treblinka.) If memory serves me well most victims were massacred in their own homes so there might be a few areas in the capital that score very high on the list of most blood soaked square miles in history.

If you remove the violent death requirement, major cities would become candidates too: given enough time, any densely populated area will eventually clock a large number of deaths. (More so even when they're bombed as raised in Loong's answer.)