Remember air is made from atoms like nitrogen, oxygen etc.. and so they can't go into another atom.
Exactly, everywhere we go on the earth's surface, we're 'wading' and pushing ourselves through a compressed gas in effect. Air, as we know it, is simply made up from a mixtures of atoms and molecules, so the smallest thing in air would be an atom of hydrogen for example (or if we're ignoring electrons, a proton). Everything between the atoms we move through, as we can understand, is nothing. The atoms are pretty close together from our perspective, but there's a massive amount of free space in a gas compared to a solid or liquid, where the atoms are very close together.
However, the point the programme was making was not just that there's a lot of space between atoms, but there's a massive amount of space
within atoms themselves. There's a central nucleus, which is very small, and then orbiting electrons at a distance comparatively very far away from the nucleus. Of course, to us, it's a tiny distance, but to the tiny nucleus, it's very large.
When we touch something, we're just touching the electrons, not the nucleus. Hence some of the 'floating on your chair' comments above. Why do we not simply move through our chair if so much of it, and us is free space? The answer lies in repulsion (no, not the problem a lot of us geeky chaps have with the ladies, but of electrons). Electrons, that essentially define the size of the atom, are negatively charged. As we all know, similar charges repel, and this applies on an atomic scale too. On the atomic scale, this is a very strong force. It is one of the defining characteristics of our universe, one of the major forces of nature as we know it. Bring atoms close together, and the electrons repel each other. Of course, this isn't the only thing that can happen, or we wouldn't have any chemical reactions (essentially just a movement of electrons!), and the universe wouldn't exist as it does. So, we're sitting on our chair because the repulsive force of atoms (because of their electrons) is balancing the force of gravity (an interesting and slightly perplexing force in our universe for other reasons).
Now as I said, atoms, nuclei and electrons don't always just repel. A really important discovery in physics (and the sciences in general) came from Ernest Rutherford, a name you may be familiar with. In probably his most famous experiment, he shed light on the internal structure of the atom by firing alpha particles (a He nucleus) at a sheet of gold foil. Most of the nuclei passed straight through, but some were scattered by the gold atom's nuclei. This is a classic experiment, but still left many questions about atoms that have since been answered. If you want to read more about it,
this page isn't bad.
Nowadays nuclear physics is more concerned with not the first components of nuclei (neutrons and protons), not just the components of neutrons and protons (known as quarks - not the bar in DS9), but the components of the components of the components of nuclei! My knowledge here is sparse as I do chemistry, and that's fairly detailed physics. I'm pretty sure there are some physics graduates on OcUK who could answer any questions on that.
