Things are not empty space. Our classical intuition fails at the quantum level.

The "volume" we usually ascribe to an atom is precisely the volume in which its electrons have a non-zero position expectation value, i.e. in which it is possible that we, if we look, find an electron.

Matter does not pass through other matter mainly due to the [Pauli exclusion principle](http://en.wikipedia.org/wiki/Pauli_exclusion_principle) and due to the electromagnetic repulsion of the electrons. The closer you bring two atoms, i.e. the more the areas of non-zero expectation for their electrons overlap, the stronger will the repulsion due to the Pauli principle be, since it can never happen that two electrons possess exactly the same probability to be found in an extent of space.

The idea that atoms are mostly "empty space" is, from a quantum viewpoint, nonsense. The volume of an atom *is* filled by the wavefunctions of its electrons, or, from a QFT viewpoint, there is a localized excitation of the electron field in that region of space, which are both very different from the "empty" *vacuum* state.