Pauli Exclusion Principle!

Electrons guard their territory around the nucleus and don't like to share with other electrons (unless opposite spin), and there's only so much space around the nucleus so other electrons are ultimately repelled. If an orbital does have openings, and the charge on the nucleus is strong enough to accept the additional electronic charge, but not too strong that it rips the electron from the other atom, then a bond may form.

Carbon (most plastic stuff and most of our bodily components are made primarily of carbon) only has four openings in it's outermost orbital (its valence shell). Those are typically taken up by hydrogen's electron (the electron is shared between carbon and hydrogen since both nuclei are strongly attracted to that electron's charge), or other carbons (same deal with sharing, i.e. covalent bonding). Other possible elements bonded to carbon include nitrogen, oxygen, sulfur, silicon, and occasionally others. These elements are all rather small, so only bind with a few other small elements, and bind strongly (relatively speaking - you can often break bonds easily using mechanical force to tear/bend plastic or whatever, but when it comes to forming new bonds you are working on a much smaller scale, using individual atoms).

We DO form hydrogen bonds all the time, tho. This is when there is a polar interaction between the positively charged hydrogen nucleus and the negative charge of electrons on another electronegative atom (e.g. oxygen, nitrogen, etc), but the electrons don't join hydrogen's orbitals - it's just a polar interaction. This is why/how water sticks to our skin.

This was a question meant for a chemist :)