What is the distance between, say, a cup of coffee and the table it rests on?
What is the distance between two touching hands?
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.
This very much depends on how you define the measurement. The simplest answer would be that the distance is zero since they are touching, but I assume you are making reference to the electromagnetic forces/Pauli exclusion principle that keep the electron clouds of the molecules and atoms from actually intersecting. However, since the electrons are in clouds and not at any specific positions, one cannot easily define how far apart one cloud is from another. Also, when objects touch, bonds are formed between molecules and electron clouds partially or fully merge. The other option is to specify the distance between the closest two nuclei, which is also tough to do because some nuclei will protrude into the other material. At an atomic level, this is not a very meaningful question.
So the summary is:
Macroscopically: The distance is zero
Molecularly: The measurement is hard to define, but Van der Waals forces, etc include the partial to full merging of molecular electron clouds. This would incline me to say zero again.
Nuclear: There is a distance between two "touching" atoms when measured from the nucleus, but for objects like hands and coffee cups and tables, it is difficult to speculate on the distance between surfaces. Atoms from one object will protrude past the mean surface of the other object. Cold welding, inter-molecular forces, and inclusion of different elements also makes it hard to determine at this level where one object ends and the next begins and what the mean distance between the nuclei of the outer surfaces of both objects is.
That all said, the only answer I can think of that even amounts to a hill of beans is that the distance between neutral touching objects is zero.