I've read that they required draw forces of up to 145 lbs, but this sounds insane.

The draw weights on longbows found on the wreck of the Mary Rose were estimated to have varied from 100 to 180 pounds. There are plenty of Chinese bows from the Qing period in museums with measured draw weights of 140 - 170 pounds. There are two bows belonging to the Kangxi emperor at the Palace Museum, with draw weights of 100 and 147 pounds. Lots of Ottoman bows at the Topkapi Palance armory in Turkey have measured draw weights of 160 pounds or more. Plenty of Mughal bows in India have been measured at 140 - 160 pounds.

So it's not a longbow phenomenon exclusively - there have been many different bow types in history with high draw weights, pretty much from every place in the world where warbows were a big part of combat.

There are a couple things to consider. Just because we find high draw weight bows doesn't mean they were the only kind of bow used, or even that they were the commonest. Not every soldier was a dedicated archer, but lots of soldiers were given bows to use at range before the combat got closer. The average draw weight was probably much less, because drawing a powerful bow takes many years of practice, and most soldiers didn't have that.

There's a report from 1736 of the Hangzhou garrison in China, which consisted of 3200 troops. Of these, about 2200 were using bows with draws between 80 - 133 pounds, while 80 were using bows in the 147 - 173 pound range. The rest were somewhere between those two extremes. The Chinese graded bows according to "strength", so those 80 - 133 pound bows were graded "strength 6 to strength 10", while the 147 - 173 pound range was graded "strength 11 to strength 13".

Armies had graded systems, so all soldiers weren't using the same bows, they were adapted to the soldier's strength and training. There are instruction manuals on how to choose the proper bow for your strength, and how to progress to high strength bows as you get better and stronger.

In this context, it's not surprising if longbowmen, who were highly trained professionals introduced to archery since childhood, used high draw weight bows.

Another thing to remember is that how the draw weight is felt by the archer also varies between bows. Longbows are actually pretty easy to pull, because they aren't front loaded. At a 20' pull, a longbow only reaches about 45% of its maximum draw weight, so most of the draw weight comes at the end of the draw, when you've already done the "shoulder roll" to stabilize your shoulder.

Contrast this to compound bows like the Chinese or Indian or Manchu or Ottoman bows, where a 20' pull represents around 75% of the maximum draw weight. These bows are much harder to handle than a longbow and you're much more likely to injure your shoulder on the early draw before your shoulder joint reaches optimum configuration to take the strain. But on the other hand, they make horseback shooting much easier, because even a short draw has the power to drive a mean arrow.

Generally speaking, at the 60 pound "competitive" shooting range, you are in toy bow category so far as warbows are concerned. At around 80 pounds, you start getting into useful territory, with sufficient range for bow fighting and sufficient penetration for medieval armor. The vast majority of military bows all across the world start at 80 pounds and then go up.

You'd expect soldiers who only occasionally use bows to stick to the 80-100 pound range, since it's easiest for those without a lot of training. It's high enough to do good damage, making it a worthwhile weapon. When you get to professional archers who were trained in warbows since childhood, 120 to 160 pounds is not unusual. Regardless of how it may feel to modern archers, these are measured draws from actual historical warbows in museums across the world, including Europe, China, India, the Ottoman Empire, etc. Such high draws may not have been very common, certainly the average soldier didn't use them. But there were enough elite archers who did that such bows were manufactured in significant numbers.

Are we really to believe that England had thousands of these longbowmen who could casually row a 145 lb dumbbell? That simply sounds insane by modern strength standards.

There is a bit of contention over exactly how many soldiers could shoot very high strength bows, as well as whether they could really do so "casually". The Mary Rose was the English flagship and it's crew may not have been representative of the rest of the army. It also possibly represents the peak of longbow archery, since this was a point where armor was growing thicker and thicker to counter newly introduced firearms. After the Mary Rose sank longbow archery seems to have declined rapidly until it was completely abandoned by the english army less than 50 years later. In 1570's, pamphleteer Barnabe Rich estimated that if 1000 english longbowmen were mustered from two counties, within a week only 100 of them would be able to shoot an arrow beyond 200 paces, with the average range being between 180-200 paces. Humfrey Barwick, an English mercenary from the later 16th century and a major critic of archery claimed that he frequently saw archers grow tired after a few shots and stop drawing their arrows all the way back, or that after weeks of sleeping in the cold and eating poorly while on campaign they were often too weak to draw their bows at all.