Pauli exclusion holds neutron stars and atomic nuclei apart. ie. much denser than atomic contact.
Even with the clouds overlapping, I think it's mostly electromagnetic. They are too sparse for exclusion to be significant.
To get any deeper, we would need someone who understands the source and mechanism of exclusion.
The Pauli exclusion principle applies to all fermions, including both electrons and nucleons. The PEP for nucleons is what keeps neutron stars from collapsing (normally). But the PEP for electrons keeps electron clouds from overlapping (much).
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: