I have yet to climb a 20ker but I climbed Kilimanjaro which is about 19K and change. The next highest mountain I've climbed was Whitney at just under 15K. I believe it's the highest in the contiguous US.

Both climbs didn't really require any technical training since both are pretty much just long, steep hikes so I can't really speak to the difficulty of something like Everest where oxygen tanks, ice axes, etc. are needed. I did use crampons for Whitney though as the terrain closer to the summit can be a bit icy and dangerous.

Training-wise I was in fairly good cardio condition when I decided to do Kili. I had been hiking 3-5 miles once every week and running that distance about every other day. Honestly, I think anyone who is in solid walking condition can summit Kili if they do it slowly and have the mental fortitude for it. The only real issue is the altitude. No one can predict how they'll handle being that high up and there's really no way to prep for it besides to actually do it.

Whitney was a different story. I did it a few years after Kili and had packed on some weight and was no where near in the same cardio shape as I was for Kili. But I thought eh, I've been at 19K, I can do 15K no problem. Nope nope nope. And the biggest obstacle: We did it as a day hike. 20+ miles roundtrip. 14.5K ft. Stupid of me to be so underprepared for something like that. We made it but my legs were destroyed afterwards.

Planning on doing the cactus to clouds hike in palm springs sometime soon, you can bet I'm in much better shape now than I've ever been. Plenty of trail running and I've been trying to hit at least 1 10ker ever 3 or so weeks.

If you only have access to a gym and no hills or trails to practice on, stairmaster is the shit. It has a rep as a sort of machine for bitches but I maintain that it's the most difficult and best cardio machine in any gym. Also, make sure to train going downstairs too. Summitting a mountain is tough, but coming down the mountain is much harder.