My husband is one of those in the camp of: If it was built and it can be climbed, well, we are doing it!!! So, I normally am part of the climb, too.
As others describe, there are little (not large at all) alcoves where one can sort of scoot into and allow a little room for someone to pass, and the alcoves usually have tiny little windows. But, yes, near the top the ceiling is definitely curved. I'm just under 6 feet tall, so I was doing a lot of leaning, as was my spouse. When you get to the very top of the stairs, the stairs are pretty narrow at the point, and you would be REALLY leaning. Then, I'd have to pull photos to refresh my memory, but the very last part, I remember holding on to a huge rope going up maybe 2 dozen small wood stairs....but once you make it to that point, you would probably climb ANYTHING to just finish!!!! As another poster said, it is extremely crowded outside the dome, but the view is oh so worth it. And, it is fun to see the spikes that used to be used for candles (think about the pour souls who had to climb out on them to light them pre-electricity days).
Would I do it again, if I had not already climbed it? Yes!! And, on one of the platform type things almost all the way up, one can walk out to where some of the statues or carved figures (whatever they are called) that one looks up and sees from the square (that look so small from on the ground) and see just how BIG they are.
We were there in early April, the year of the beatification of John Paul. I don't know if it was extra crowded because of the activities related to that, but we climbed late in the week, after all the festivities were finished (so many had already left Rome).
Various people are bothered by small spaces in various ways. Perhaps there is a video someone may have posted on YouTube of doing the climb. I'll sometimes check YouTube, as it seems there is a video of anything one can imagine in life :)