I'm not a physicist, but when I looked into this, I found this well-written article:
The Stability of Matter: From Atoms to Stars
It goes into lots of detail of what's happening with a single hydrogen atom, then a large atom, then bulk matter. It doesn't require quantum physics knowledge from a reader, but it does require mathematical maturity, and isn't easy reading.
The short of it is that you're right, the Pauli exclusion principle is more important than electrostatic repulsion.
Thanks! I love docs like these, that take the a broad approach.
I thought this video was a really good question dissolving by Richard Feynman. But it's in 240p! Nobody likes watching 240p videos. So I transcribed it. (Edit: That was in jest. The real reasons are because I thought I could get more exposure this way, and because a lot of people appreciate transcripts. Also, Paul Graham speculates that the written word is universally superior than the spoken word for the purpose of ideas.) I was going to post it as a rationality quote, but the transcript was sufficiently long that I think it warrants a discussion post instead.
Here you go: