Interatomic potential is usually described as consisting of two parts: one attractive and one repulsive, where the repulsive one prevails at short distances.
The repulsive part is typically explained in terms of the Pauli repulsion principle, by saying that in order to allow for the electron clouds of the two atoms to overlap, electrons from filled subshells will have to be promoted to higher levels, which costs energy.
However, in the case of two hydrogen atoms, there are no filled subshells. First of all, for two hydrogen atoms where the electrons are in different spin states, there is not going to be any Pauli repulsion, so the repulsion must have a different origin. Of course there is some electrostatic repulsion between the electrons, but at very small distances I guess it's the repulsion between the nuclei that prevails.
Consider now the case where the two electrons have the same spin state. One way to overcome Pauli repulsion is then of course the promotion of one electron from 1s to 2s. Perhaps this promotion should be viewed in terms of wave functions for Helium? However, would it not also be possible to instead alter the spin state of one electron? If they are both in "spin up" state, actually this would release energy. I know such a transition is forbidden under E1 (although allowed under e.g. M1) but since we consider two interacting atoms, I reckon the transition could occur through collision.
If the two electrons are both in "spin down" perhaps its more complicated. Could one electron be promoted to "spin up" (still in the 1s orbital) by collision between the atoms?