I'm sure most soldiers had unbelievable stamina from training every day of their lives

The absolute and vast majority of soldiers throughout history had nothing close to that level of training. Even among warrior elites you would have to look long and hard to find something like that.

Fighting for hours and hours with metal weapons you have to thrust and lift and throw, while wearing warm and heavy clothing sounds super-human to me, so how were they able to manage that?

It is superhuman. The answer is simple, the didn't do that. It's impossible to give a general answer for all time periods and regions but generally speaking you can't keep fighting for more than a few minutes before your effectiveness starts dropping off quickly. To combat that a wide variety of tactics have been employed throughout history, depending on situation and the quality of troops at your disposal.

A roman imperial army would (probably) rotate ranks, effectively letting the first line fighters take a step back to that the second line becomes the first. It's effective as it allows you to keep pressure on the enemy formation but requires a great deal of training and discipline to carry out.

A medieval levy would likely clash and withdraw to "dress the line" (basically modern day NCOs shouting at people to get back in formation, get the wounded out of the way and get fresh fighters to step into the first rank), then they'd clash again and repeat until one side broke. Ideally you wanted to be quick about this so you could hit the enemy formation before they had finished dressing their line, hopefully routing them.

Either way you look at it a few things remain more or less constant. No one can fight for extended periods of time without becoming exhausted. No soldier will willingly put himself if a situation where death or serious injury are the only two escapes from the first rank. Given this, you need a system that works and in very broad terms that system is reserves. Whether or not you arrange it with reserves internally in a unit (as in the roman case) or by fresh troops being rotated in during a lull in combat the end result is the same, fresh units step into the first rank every so often.

I'm wondering whether an ancient or medieval battle really meant non-stop, hand-to-hand fighting for all soldiers at the same time.

Not it almost certainly didn't unless the situations were extremely dire. When that kind of situation does arise, as it did for the roman legionnaires at Cannae for example, the result is generally a slaughter.