What is the distance between, say, a cup of coffee and the table it rests on?
What is the distance between two touching hands?
Answer
This answer I once gave for What does it mean for two objects to "touch"? discusses what touching even means. It's not a direct answer to your question, but I think it may help you view the issue in a different way. Warning: It's one of my long, talky answers that some people love and others hate. The physics in it is accurate (and for many folks, unexpected) in any case.
The specific answer to your question is that the most fundamental distance between two touching objects is determined by Pauli exclusion surfaces between electrons in the touched and touching objects, with the surfaces being where there is zero probability of finding electrons from either of the objects. Thus how "close" the objects are depends on what level of normalized probability of finding either electron in the exclusion pair you are willing to tolerate. E.g., for some specific set of nearby atoms, "1%" gives one (very short, sub-Angstrom) distance, while "5%" gives another, somewhat larger distance.
Oddly, that also means that the simplest answer is that the objects really do "touch", specifically at the surface of zero probability due to Pauli exclusion.
There are other modifiers of course, such as thermal noise that bounces these surfaces apart at very high frequencies and so give various types of averaged distances. The deeper physics of actual repulsion always, for ordinary matter, goes back to those Pauli exclusion surfaces between individual pairs of electrons in the touched and touching objects.
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