^If particles didn't interact? Depends on what particles you mean. If molecules and atoms didn't interact, you'd have a gas. If protons and electrons didn't interact, you'd have a cloud of protons, electrons, and neutrons (and some other stuff), but not even atoms. You can't really call it a gas if there aren't any atoms. If quarks didn't interact, we'd have a cloud of quarks and electrons because quarks bonded together make up protons and neutrons. (And a lot of smaller and larger stuff which doesn't interact in the way we're used to.) There may be stuff even smaller than that, but we can't see it yet. Forgive my fuzzy particle physics, though.
In chemistry, we say that atoms are "bonded" when their electron clouds overlap to a significant degree... That degree, however, is mostly just convention.
Things that aren't bonded together, like your butt and your chair, are even more tenuous, but you do feel the chair underneath you and that's because the atoms of the chair are close enough to interact with your skin, which interacts with your nerves...
Personally I think we should define "touching" more in the biological sense because it's actually a lot more intuitive. I mean, if you feel something, you're touching it. xD;
Anyway, if you're only in a Newtonian mechanics class, none of this really matters. You might as well consider objects to have 0 distance between them, as you can't calculate the distance using the tools you have because a) you don't have enough significant figures, and b) quantum mechanics starts to play a big role. A lot of physics is about ignoring the forces that are too small to matter. That's what I learned in Modern Physics, anyway. =P Taking Quantum in the fall.
EDIT: Anyway, to begin to answer the OP's questions... The only "weird rounding" going on is the limit of how weak a force our nerves react to, and how much our brain is willing to focus on that sensation. Some people have more sensitive fingers than others. You don't need to be "touching" anything in the strong sense you want to feel it... just close enough for the object's electron cloud to interact with your electron cloud. And "0 distance between two objects" would technically make them the same object. o_o Since matter can't overlap and all. In the case of two nuclei, this would mean nuclear fusion...
But if you said to a physicist that Object A and Object B were at 0 distance from each other, the physicist would usually assume that Object A and Object B are two labels for the same object, if the distance refers to the center of mass, or that they are two parts of the same mass, if the labels refer to the "boundary"... whatever that means at 0 distance. Usually we label from the center of mass, so 0 distance doesn't have the meaning I think you want with that system.
In chemistry, we say that atoms are "bonded" when their electron clouds overlap to a significant degree... That degree, however, is mostly just convention.
Things that aren't bonded together, like your butt and your chair, are even more tenuous, but you do feel the chair underneath you and that's because the atoms of the chair are close enough to interact with your skin, which interacts with your nerves...
Personally I think we should define "touching" more in the biological sense because it's actually a lot more intuitive. I mean, if you feel something, you're touching it. xD;
Anyway, if you're only in a Newtonian mechanics class, none of this really matters. You might as well consider objects to have 0 distance between them, as you can't calculate the distance using the tools you have because a) you don't have enough significant figures, and b) quantum mechanics starts to play a big role. A lot of physics is about ignoring the forces that are too small to matter. That's what I learned in Modern Physics, anyway. =P Taking Quantum in the fall.
EDIT: Anyway, to begin to answer the OP's questions... The only "weird rounding" going on is the limit of how weak a force our nerves react to, and how much our brain is willing to focus on that sensation. Some people have more sensitive fingers than others. You don't need to be "touching" anything in the strong sense you want to feel it... just close enough for the object's electron cloud to interact with your electron cloud. And "0 distance between two objects" would technically make them the same object. o_o Since matter can't overlap and all. In the case of two nuclei, this would mean nuclear fusion...
But if you said to a physicist that Object A and Object B were at 0 distance from each other, the physicist would usually assume that Object A and Object B are two labels for the same object, if the distance refers to the center of mass, or that they are two parts of the same mass, if the labels refer to the "boundary"... whatever that means at 0 distance. Usually we label from the center of mass, so 0 distance doesn't have the meaning I think you want with that system.