Both the Pauli exclusion principle and electrostatic repulsion contribute. There is a brief discussion of this on Wikipedia, which cites the work of Freeman Dyson.
A more rigorous proof was provided in 1967 by Freeman Dyson and Andrew Lenard, who considered the balance of attractive (electron-nuclear) and repulsive (electron-electron and nuclear-nuclear) forces and showed that ordinary matter would collapse and occupy a much smaller volume without the Pauli principle.[6] The consequence of the Pauli principle here is that electrons of the same spin are kept apart by a repulsive exchange interaction, which is a short-range effect, acting simultaneously with the long-range electrostatic or coulombic force. This effect is partly responsible for the everyday observation in the macroscopic world that two solid objects cannot be in the same place in the same time.
I guess Feynman includes the Pauli principle as electric force. Remember, he got a Nobel prize for this stuff.
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: