How was Israel able to build a powerful military within days of the formation of the state?

The Israel had a number of paramilitary groups which had been training, stockpiling arms long before the 1948 war, many Jews joined the British forces during ww2 to gain military experience. The Zionists had created a shadow army and had stockpiled weapons for a long time before 1948.

The Local Arab forces were very disorganised divided and had little weaponry by comparison, and their supporters in other Arab states were reluctant to effectively arm them no lest because they were keen to suppress any possible Palestinian state and annex land the could form that state.

The Arab intervention forces were poorly organised and equipped in General and very divided politically. The Arab Legion was pretty well equipped and organised (though not that well supplied with ammunition that had not stockpiled well and the blockade of resupply was much more effective on Arab forces than Zionist forces, who were better able to circumvent the arms embargo during the war and cease fires) Jordan sought to reach an understanding with the Zionist leadership and was primarily focused on annexing much or the proposed Palestinian state rather than attacking the new state of Israel the only real disagreement being Jerusalem. Syrians were involved in a limited land Grab, the Lebanese made a demonstration not an attack, and Iraqi forces were troublemakers shipped off rather than supported army force.

The Syrians used extremely dated pre ww2 French tanks of dubious value. The Israeli forces were not fighting well developed and equipped armies. The Arab states were also fairly new, badly equipped with obsolete weapons operating in many cases a long distances from the logistical bases.

Actually she wasn't able - the Israeli army of that time was quite far from being mighty. It was a more or less hodge-podge assembly of what they could assemble during the British rule and what they could import just after the British left. The thing is (as other replies pointed out) that the invading Arab armies (with the exception of the Arab Legion from Transjordan) weren't any more mighty, even less so.

During the British era prior to the Independence there was a ban on weapons so what they could smuggle in and hide from the Brits were quite limited in quantity and quality. But even that time there were local conflicts between Arabs and Jews and it was clear that the moment the Brits leave there will be a wide-scale open conflict. So they built up paramilitary militias (like Haganah) which later during the war served as the core of the new army (other replies give a good view on this) and tried to arm themselves: smuggle in weapons and also manufacture them while hiding all of these from the Brits. E.g. they exported civilian airplanes to use in agriculture then secretly converted them to military use - once the Brits left these converted civilian airplanes formed the first batch of the new Israeli Airforce. They manufactured weapons, copying the British arms (like the Stern machinegun https://en.wikipedia.org/wiki/Sten#Palestininian_Mandate_Stens ) or even inventing new types like the Davidka mortar which wasn't really of any use as a mortar but was loud and scary which made it quite useful in scaring the enemy (https://en.wikipedia.org/wiki/Davidka). Another preparation for the war was that the Jewish Agency tried to buy up weapons they could get in Europe and in the US. They could buy up a variety of weapons, usually outdated or inferior versions of more modern weapons (like the Czechoslovak Avia S-199 https://en.wikipedia.org/wiki/Avia_S-199) but as the enemy wasn't equiped any better it gave the Israelis quite the edge during the war. The Jewish Agency made a deal with Chechoslovakia, purchased arms from them and had them store them in Chechoslovakia. They smuggled in what they could during the British rule and started to transport the rest (like the airplanes) once the British left (https://en.wikipedia.org/wiki/Arms_shipments_from_Czechoslovakia_to_Israel_1947%E2%80%9349). This was called Operation Balak (https://en.wikipedia.org/wiki/Operation_Balak)

As a matter of fact, the 1948 war actually started on November 30th, 1947 - the day after the UN Partition Resolution, as the Arabs vowed not to accept it. The first phase of the war pitted Palestine Arab irregular warbands against Jewish paramilitary formations - the mainstream Haganah, the more nationalist Irgun, and the really really radical Lehi. The Arabs were largely beaten in this phase. During the second phase, the armies of five regular Arab states invaded the newly formed State of Israel and by and large met with defeat. (The only one of them to acquit itself well was the Transjordanian Arab Legion, a British-trained and partially British-officered unit).

The main brunt of the first phase of the fighting was borne by the Haganah. Where did it come from? Well, for the duration of the British Mandate in Palestine, the Jews were building, with British sanction, the institutions of a proto-state, usually called in later historiography "medina shebaderech", literally meaning Hebrew for "a state on its way". This included democratic elections of quasi-executive and quasi-legislative councils, trade unions, healthcare institutions etc. (Wikipedia is very brief on this - but still worth a look). It was also building a paramilitary capacity, known as the Haganah - unlike the rest of the institutions, this was mostly against British wishes. (Haganah-British relations are a somewhat complicated subject, which I haven't time to treat here fully). Haganah was also practicing a form of "voluntary conscription" - young men were expected by their society to join its training for a year; those who declined to do so were socially frowned upon.

So, when the newborn Israel needed an army, it had the nucleus of one - Haganah. This is the story in brief.

One source which treats this from an interesting angle is a chapter in the book Supreme Command by E.Cohen.