What repulsion keeps non-hydrogen atoms at a distance: between inner shells or between nuclei?
I found a nice figure and the relevant statement in a paper by Frenking and Krapp (Unicorns in the world of chemical bonding models, 2006, https://doi.org/10.1002/jcc.20543):

The crucial term which is responsible for repulsive interactions in chemical bonds except in two‐electron systems such as H2 is the Pauli repulsion. The three terms (a) quasiclassical electrostatic interaction, (b) resonance, and (c) Pauli repulsion give a complete picture of the interatomic interactions which yield a chemical bond.

The figure shows how the distance-dependent repulsion for one scenario including the Pauli repulsion and two scenarios ignoring the Pauli repulsion; the latter do not show the drastic increase in repulsion that is observed.

Source: https://onlinelibrary.wiley.com/cms/attachment/2b2dc015-ca3c-4fd3-b51c-f768a6a924b5/mfig003.jpg
The entire essay is great, connecting the picture of bonding that bench chemists have with the developments of theory and computation in the 90 years since quantum mechanics was first applied to chemistry.