Particles are separated into fermions and bosons.
Fermions, such as the electron, proton, neutrons, neutrino, have a quantum number with half spin.
Bosons such as the photon, and some atoms, have an integer spin.
Fermions cannot collect together and have the same quantum state. This gives rise to the chemical nature of atoms and the quantum shells, which I am sure you have heard about. Two electrons, each having an opposite spin from the other can occupy a shell. The next electron that an atom possesses must reside in another shell, and so on.
This is the basis behind the Pauli exclusion principle, as proposed in the first part of the 20th century.
you said,
My understanding of what allows me to touch, say, a desk is that the desk electrons become very near the electrons on my hand and the electric force repels them.
Certainly that is true. But what if you press down more and more? Can you overcome the electromagnetic repulsion of the electrons, and push your hand through the desk?
Well no. The electrons like a certain size of shell and resist extra pressure being put upon them with what is called degenerency pressure, which is a result of the Pauli Exclusion principle. Your extra pressure is trying to push electrons together so that more than 2 will fill up a shell. The more the pressure the more the degenerant pressure. Ordinary matter that one deals with has some degenerant pressure but, for the most part electromagmagnetic forces dominate.
Planets, with a much great interior pressure, and no great internal heat source to provide thermal pressure, are held up, as you can now imagine by a good deal of degenerative pressure from the electrons.
From this graph, from the site listed, gives an indication of the repulsive quality at small distances, for Argon, which which even if attracted to one another at first, the repulsion due to Pauli exclusion dramatically increases as they become very close to one another.
http://www.quora.com/Physics/Is-it-...t-explain-why-I-do-not-fall-through-the-floor
For some celestial objects, the pressure is so great, that the only repulsion that is holding the thing up is for all the electrons to become degenerative and you have a white dwarf.
If the pressure is even greater, the electrons are forced to merge with the protons in the nucleus to form neutrons, and the neutrons will provide the degenerative pressure to hold the object up in what is called a neutron star.
So here, even if the strong force is only attractive, and increases with distance, that would be the case for only the quarks exchanging gluons within each individual neutron. Two neutrons trying to merge under great pressure, or energy, will behave similar to the electrons under pressure, and Pauli Exclusion will again come into play, but this time for the neutrons.
That probably is too simple of an explanation of what goes on, but even if it does not satisfy a good answer, you will read as much as you can to acquire a better understanding. Perhaps my low percentage of knowledge of nuclear interactions has helped you out some.
Cherios.