Weights were not scientifically plausible until 600 years later. (Quote from www.precisa.co.uk/the-history-of-the-weighing-scales) Even Leonardo da Vinci failed to invent a reliable weights system. If scales were 99% accurate, on the third instance of weighing after the Royal standard, the error could be as much as 3%.

Ancient pounds appear to be around 330 to 700 grams. In England it was 330g, then 375g (The Troy pound started in early renaissance, Henry VIII) and later became 453g. Weight was based on grains of wheat, 7860 grains of wheat was a pound, it was the most accurate way that anybody had figured out until 1770.

The times of Robin Hood were full of banditry and local feuds. The King was in prison in Germany for 3 years at that time for a ransom of 150.000 marks, and was otherwise absent for the crusades. The laws and weights relied on big centralized governance when corruption was notorious.

In France, a pound weighed between 380g and 500g, depending on the province.The roman pound was 330g, the Hapsburgh pfund was 560 grams, and the Scottish pound was about 700g.

https://en.wikipedia.org/wiki/Anglo-Saxon_pou

The pound was a unit of account in Anglo-Saxon England, equal to 240 silver pennies and equivalent to one pound weight of silver. So, 300 pounds, was 300 pounds (136kg) in silver

(Disclaimer: I am using very, very round numbers for my estimates.)

The description of hundreds of pounds (ranging from 100 to 800) seems to come from some of the earliest surviving Robin Hood tales, printed in the early 1500s but probably based on earlier works. So the writer was probably not thinking of paper money but may have been more familiar with some slightly lighter coins--but still significantly heavy.

(Side voyage into Tudor coinage from the mid-1500s: 1 Troy pound of silver = 60 shillings. With that conversion: 100 pounds Sterling ~ 25 lbs modern weight, 300 pounds Sterling ~ 80 lbs modern weight, 800 pounds Sterling ~ 215 lbs modern weight. Any earlier than mid-1500s and it's heavier than that.)

(You haven't mentioned the volume, but using a converter: 25 lbs of silver is ~1/3 gallon, 80 lbs ~ 1 gallon, 214 lbs ~ 2.5 gallons.)

Likewise, the value of 300 pounds (money) would have been somewhere in between modern and early middle ages, but still pretty significant. Using a relative value calculator, in 1550 we have 300 pounds ~ 100,000 pounds of value or over ~3 million pounds relative income. In 1270 (earliest year available) we have 300 pounds ~ 300,000 pounds of value or ~10 million pounds relative income.

The money is (depending on version) either to buy a large plot of land with a lot of animals or to buy a pardon for killing two men (it's complicated). So the value actually seems reasonable.

This seems like the historical briefcase full of money. The numbers are on the edge of plausibility, physically, but the actions of the characters (wandering around alone/with strangers while carrying this much money) seem driven more by narrative requirements than common sense.

In that era, 300 pounds weighed 300 pounds, but it's a different pound.

Your example of Robin Hood fixes the time period during the reign of King Richard about 1175, though we must remember this is a legend. This is just after the chaotic reign of King Stephen when the centralized minting of coins broke down. His successor, King Henry II, reintroduced the royal mint system. Rather than pure silver he began striking coins from 92.5% silver, sterling silver which is harder than pure silver, thus pound sterling.

In that era, 300 pounds weighed 300 pounds. No matter the coinage, it's 300 pounds of silver; 240 silver pence weighed and was a pound. This makes checking the value of a bag of coins very easy regardless of whether they're shaved or not, it was weighed.

However, their pound was not our pound. In King Richard's time, a pound of money would be based on the Tower, Saxon, or moneyers' pound of 12 Tower ounces. This was roughly 350 grams or roughly 75% the weight of a modern pound. The modern 16 oz pound is the London pound.

So Robin Hood's 300 12 oz pounds of coins would weigh roughly 225 modern 16 oz pounds (around 100 kg).

many medieval pictures of people riding depict horses which seem very small to me

Careful, medieval art was not concerned with accuracy. They would regularly depict any scale or perspective that fit whatever they were trying to say with the art.

As for carrying it on a medieval horse, this is dead weight hanging down with a low center of gravity, not live weight sitting high above the horse. 20% of the horse's weight is a modern humane consideration for a modern light riding horse. A medieval horse would be less differentiated and expected to carry loads as well as a rider, or the rider could walk along side while the horse carried their load.

Where there were any gold coins minted in England during the middle ages?

Yes, but gold was worth roughly 20 times more than silver. If you made a gold penny the same value as a silver penny it would be tiny and slip through gaps in your bags. To keep gold pennies a reasonable size they were worth 20 silver pennies.

See also

• Pounds, shillings, and pence: a history of English coinage