How do you get to 0?

This. If you think about an object, like say a chair. It's mostly empty space. The particles only take a very small fraction of the vacuum. They interact though: They stick together in such a way it gets the chape of a chair. Even your own body is 99% or so vacuum and a small % is matter (you can even debate it is 0%, since particles don't really exist since they're all wavelike, like light, but that would go too far).
When you push something, you come so very close to the object, the particles of your hand are interacting with the particles of the object, simple electromagnetic forces, which get so great at some point they move the whole object, so your hand never touches the object.
If you keep pushing very, VERY hard, untill you overcome the E/M force, and the atoms collide, I guess you'll trigger nuclear fission, but the force would be so great it'll never be possible by human force. (not so sure about this fission part though ^^)

Actually, technically your body is 99.999% empty space. The nucleus of an atom is 10^-13 CM wide, while the electron cloud is 10^-8 cm. In other words, an atom is 5 times the size of the nucleus.

You're not actually sitting in your chair right now. The forces of repulsion created when electrons get close is causing you to hover very slightly above your chair.

There's only one time when the nuclei of atoms touch. It's called fusion