The short version: It's the Pauli Exclusion Principle.
6 paragraphs from the end of the article they actually get to the point.
Yeah they really make you work for this answer.
Electrostatic repulsion is also an indispensable component, so the answer obviously isn’t that short.
To quote the article;
"The Pauli Exclusion Principle doesn’t only explain why matter is solid, but also why it occupies the amount of space that it does. Again: it isn’t just the uncertainty principle and electrostatic repulsion that’s responsible for volume; if matter were made of bosons, it wouldn’t occupy space in the same fashion that it does when it’s made of fermions"
Which, in turn, is a consequence of the spin-statistics theorem.
As for why the spin-statistics theorem is true, the answer is that, in a sense, we do not really know. This is because, although we have rigorous mathematical proofs that it is true, they rely on arguments that are very technical in nature, so they provide no real intuitive insight into why the theorem is true. (This theorem is actually really notorious for this; people have been trying for a long time to improve on the situation, but have yet to succeed in coming up with a satisfactory elementary proof of it.)
Thank you!
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