Some considerations. First of all, the physical world doesn't seem to be continuous or infinitely divisible. Most physicists these days seem pretty sure of the Planck length and Planck time as a limit. There is the holographic principle, the Bekenstein bound, and all that. There are some good philosophical arguments that the world is discrete or not infinitely divisible that I wont get into here.
However, if matter were infinitely divisible and continuous, and for example, you had two perfect spheres touching, it seems to me that true touching would involve a point on one sphere and a point on the other occupying the same exact space. There must be zero distance between the two points, after all. Can two points with zero distance between them occupy different points in space? I don't think so.
But it seems that a whole different picture of what space and matter are has been emerging in this last century. Things like particles might not even be truly distinct objects, but rather something like excitations of a field, or waves, or some such. And waves can certainly "occupy the same space". Actually, it seems problematic to think of waves in the same medium as separate things. What would it mean for two ripples in a pond to "touch"? You might think of them as simply passing through one another, temporarily adding up, or some such. But are they really separate things, or is that just how our minds like to carve up the world?
When talking about touching, since we are dealing with distance, we are talking about space, so the nature of space or spacetime is important. I find recent thinking in loop quantum gravity, in ER=EPR, in Stephen Wolfram's causal graph, and so on, very interesting, where space isn't thought of as a big openness, an emptiness that contains things. Rather, it is a sort of network of discrete, entangled nodes that are simply connected or entangled in a certain order, with everything we call matter or empty space or whatever just amounting to certain local topological arrangements of these nodes, some of which can be knot-like or whatever, some of which can propagate like cellular automata.
Supposing something like this the above is true, what then is the touching of two material objects? I suppose it is somewhat similar to thinking about waves in a medium. What seem like two different configurations of two material objects in space, one with a distance between them and another with adjacency, are really just two different arrangements of a single network. But what then does it mean for two of these "particles" of space to be connected in the network? They are entangled. But what does that really mean? I'm not really sure. What even are the nodes and how do they interact? Are they ultimately really separate things?
I guess when we talk about different macroscale objects touching, it is just a useful-for-us way of thinking about things and carving up a world that, in-itself, perhaps isn't carved up at all. It is a high layer of abstraction, like using icons in a computer application that are like packages of nested command hierarchies of ever-simpler and more fundamental operations that would be too cumbersome to think about directly at the level of transistors and memory registers.
But you might also just think about touching as causation, or information propagation, or even as chains of contingency or layers of dependency in some sense. There has long been a suspicion, probably a pretty good one, that causation can only propagate locally, by "contact action". Scientists and philosophers have been wise, I think, to be suspicious of "spooky action-at-a-distance".
Newton was really uncomfortable with the fact that his law of gravity had nothing to say about the mechanism of this incomprehensible situation of one body seemingly influencing another across a large distance without touching. Einstein restored locality with the realization that gravity is local after all, since an object simply distorts the space it is immediately in, and then the effect propagates as warping of spacetime from one part of space to each adjacent part, and only affects the distant object indirectly when the warping reaches it and begins to affect it. It is like a wave propagating adjacently, part-to-part, in a rope rather than action-at-a-distance across a true emptiness. Space or spacetime, in this case, is "something" itself, with its own degrees of freedom.
Yes, I hear you asking about entanglement, which seems to be action-at-a-distance. But is it really? Causal influence usually seems to involve an interaction that carries information. You can't send information nonlocally with entanglement. It is simply a correlation that isn't fully understood, as of yet. In some interpretations, there isn't any nonlocal influence.
To be touched is to be influenced, to be acted upon, to be changed by something, to gain information about that something, for something of that thing to enter into you, to become part of you, for some aspect of your present state to have been partly determined by the former state of that thing, like the kinetic energy in one billiard ball passing into another. If things can be changed by other things that they "touch", can they really be thought of as truly separate and distinct in the first place? It seems to require them to be kind of permeable and ultimately without true boundaries.
And then consider that for a photon, since it travels at the speed of light, and since length contracts, approaching zero, as you approach that speed, there is no distance. It is as if the electron dropping to a lower energy level and emitting a photon, or a packet of energy, and the "distant" electron absorbing it and jumping to a higher level are, from the photon's perspective, "touching", the two electrons having zero distance between them, as if there really isn't any photon at all, but rather just the a quantum of energy passing from one electron to another in the same place. Maybe the two electrons aren't even separate in the final analysis.
In the end, it might be best to just give up on the idea that there are any separate things in the first place. Even the space particles are probably just the way in which, incomprehensibly to me, the One, whatever it is, relates itself to itself in some mind-blowing hall-of-mirrors explosion of apparent form. Wheeler and Feynman famously speculated about the
one-electron universe. Maybe it goes deeper than that.
Some in the physics community, like Nima Arkani-Hamed, are starting to talk about spacetime as emergent from something deeper. If distance isn't fundamental, then what? Can we even talk about distinct entities that could be said to touch or not touch?