I think this is a very non-trivial question and I can't give you all the answers, but I can make a few general remarks. First of all, when atoms "touch", it's really their electronic orbitals overlapping. Those interactions are, of course, governed by quantum mechanics but it is very hard to solve multi-electron atoms and complete molecules with a completely quantum mechanical treatment the way we are solving the hydrogen problem. As it turns out, one can simplify the equations significantly and for many applications still get reasonable approximations by using so called mean-field potentials.
In its most simple implementation mean field theory assumes that one can average the quantum mechanical movement of electrons around nuclei and derive a classical potential for the effective force between the nuclei. The physical reason why this works reasonable well is that the nuclei are, on average, a couple thousand times heavier than the electrons, i.e. their movement is much slower than the time scale of the electronic wave function. As a consequence it can make sense to describe molecular binding by the average distances of their nuclei and to approximate the actual dynamics of electronic orbitals with a non-linear distance and angle dependent classical force.
In general even these mean field forces depend very strongly on distance between the nuclei and on angles (and to smaller extent on the spins of the electrons), which makes for a complex multi-dimensional potential.
The radial dependence of these potentials has two general properties: at short distances the electrostatic repulsion of the nuclei (and the electrons) will be very strong and the potential has to diverge to infinity. At infinite distance, however, there is no force between the molecules and the potential is zero. At some intermediate distance there may be a potential minimum, in which case a stable chemical bond of that length can form.
I think this is where your question comes into play. One could interpret the average molecular forces between the nuclei in form of a human "touch" sensation. At a distance one wouldn't feel anything, going closer there would either be a soft repulsion that would be getting stronger very quickly, or a stickiness, maybe similar to two magnets that are attracting each other, and eventually, even closer in, there would be a rather hard surface, which one could not penetrate.
The "surfaces" of these molecular orbitals would feel incredibly slippery, because there wouldn't be any friction and there would be a constant trembling from quantum mechanical fluctuations. Depending on the combination of molecular "fingertip" and "molecular ball", the balls would either stick or constantly trying to get away. And depending on the reaction energy of the sticky ones it could be very hard to impossible to remove them without also ripping off a "finger tip" or two! In short, we would get all of the variability of chemical reactions "at hand"! It would certainly be a wonderful tool for chemists to explore chemical potential landscapes like that and I think I have heard about some folks in the virtual reality department who are working on tools like that, but I would have to find the article.