The problem is that the explainations offered tend to idealize “space” as being this large elastic blanket in which we are embedded; people visualize the blanket as expanding uniformly, and the matter within it fixed in relation to the blanket. But matter is not space, and it is perhaps a better analogy to conceptualize atoms, planets, solar systems, and even whole galaxies as lilies floating on the surface of a pond; even if the surface of the pond gets larger as more water is added, and the lilies move away from each other; the increasing water flows around them but the flowers and pads are held firmly together by their vegetable structure.
It is a property of matter (which is a form of energy that expresses itself gravitationally) that it distorts space about itself, causing other matter to be attracted and clump together–like the lilies–rather than spread out evenly; sort of like what Reagonomics did for finance moguls and their money. With large, galactic-scale structures gravity retards space from expanding. Even in larger structures, like The Local Group, gravitational attraction plays some effect, though very minor, on the expansion of local space. And even on much larger scales, enormously massive objects like The Great Attractor have a more significant effect, causing galaxies to stream toward them (but still pulling away from each other).
On more local scales, the electromagnetic force acts on atomic particles and holds together molecules, and the weak and strong nuclear forces act on subatomic particles and serve to keep nuclear particles clinging together; each is increasingly stronger, by many orders of magnitude, than its precedant, and all have much more of an effect to keep particles together than the expansion of space has in pulling them apart, in the same way that a raindrop will not “dilute” or rend apart a lilypad.
That’s a complicated answer, but the simple answer is…er, there is no simple answer. Electrons would still exist in their same shells at particular quanta from the nucleus, molecules would continue to form their present structures, et cetera, up to the point that local gravitational energy, which would be increasing, overwhelms the electromagnetic force. In the heart of the Sun, for instance, gravitationally-induced pressure prevents atoms from properly forming; instead, electrons zip back and forth, sliding between nuclei, without ever letting themselves become attached, like Jane Greer in a '40s noir film. This is called degenerate matter; it’s not quite atomic, but all the components are still intact.
Squeeze further, as with a supernova (in which the lighter mass is blown away, leaving only a superdense core) and the electrons and protons cram together to make a bunch of neutrons and some radiation which disappears into the night like an alimony payment and leaves you with neutronium, held apart only by the particles’ unwillingness to share the same quantum state with one another, like two divas attending a party while wearing the same cut and color of dress.
Add some more matter and eventually the whole party degrades into a massive orgy in which the participants become an inseverable mass of arms, legs, heads, and various other appendages; we don’t know exactly what goes on in there, and would probably be horrified if we found out (except for Bob Guccioni, who would be in a bathroom stall doing things we’d rather not think about), but we speculate that the matter would turn into a quagma, a fluid of quarks too dense for light to penetrate.
Further compression gets us to a point past which we can even guess; the best speculation is that the particles arrange themselves in a matrix in which their quantum states “interlock”, allowing them to achieve smaller and smaller gaps, until they disappear faster than a played-out teen starlett, eventually condensing to a single, unified quantum state into one giant particle. It would take an enormous amount of mass to create this level of compression–quite literally, a universe’s worth–although we hope to simulate something like it for a very, very brief instant using larger and larger particle accelerators, and hence gain a understanding of what happens in the remote corners of space and the furthest reaches of time by looking at things that happen in a space too small to imagine. How cool is that?
In any case, from what we understand right now (and while we are like the proverbial blind men feeling the different parts of the elephant, we have at least identified which smells the worst and have therefore concluded that we must be in the vicinity of an arse) the universe will not contract; the Hubble “constant” is actually a varying value that is slowly increasing over time and the energy balance of space is such that it appears that space will continue to expand away in perpetuity, until there is nothing left to look at but the formless nothingness of space accelerating away at c. And you though 2001: A Space Odyssey was boring?