What force prevents particles from penetrating other particles? I understand that what prevents objects from penetrating each other is the electromagnetic force between the electrons in the respective objects.
But if we don't have electrons, for example a proton. What force prevents a electron or a muon from going through it?
 A: First of all, for non-interacting particles, there's no reason one couldn't pass right through the other.  However electrons and protons do interact via the electromagnetic force, and an attractive force can also produce scattering (for a classical example, think of how a comet's trajectory would change if it passes close to a planet).  Also, note that not only protons but also neutrons contain both positive and negatively charged quarks (despite the neutron being neutral).  Experiments involving deep inelastic scattering of electrons off these quarks helped elucidate the structure of protons and neutrons.
A: 
What force prevents particles from penetrating other particles

This is the basic question that gave rise to the need for a different mechanics than newtonian mechanics. In different words: why does not the electron fall into the proton (and for higher Z atoms the nucleus) and charge disappear? It led to the development of quantum mechanics.
Quantum mechanics is a probabilistic theory which gives probabilities for a process to be seen, dependent on energies in the system. For example the answer to "what is the probability for the electron of the hydrogen atom to fall on the proton" the mathematics of QM answers  "zero" . There is a lower energy state, called  ground state, orbital where the electron will stay forever unless some energy enters the system. 
Forces enter in quantum mechanical equations as potentials,( electromagnetic, strong, weak,)and the solutions depend on these potentials. For example, for the orbitals of the hydrogen atom, the Schrodinger equation is solved with the 1/r potential of the classical electric field, and it  gives  the solutions which will give the probability of finding an electron  in a specific (x,y,z) around the proton, the orbitals in the link above.
The answer to "what is the probability for an electron of high energy, higher than orbital energies where it can be captured, to penetrate a proton" is "it depends on the energy" It is called proton electron scattering and QM gives us the probabilities for its interaction. With high enough energy the quarks were seen in the proton.
Penetrate also has a different meaning than classical physics.  The electron of the hydrogen atom, if its orbital has no angular momentum, has a probability of overlapping the proton , but nothing happens because of quantum mechanical stability. The same is true of heavy nuclei, but some of these are unstable, i.e. lower energy levels exist to which the neutron and proton bag can settle, and in that case there exists electron capture, a probability of capture of the electron and turn a nuclear proton to a neutron, and a transmutation to a lower Z (charge number of nuclei) nucleus by 1.
So it is a combination of energy and quantum mechanics which controls penetration, and forces act within this system( which includes conservation of various quantum numbers too that will give zero probabilities for some guesses). 
A: There is a misconception here. The proton is not a fundamental particle, it is composed by a lot of particles (if you believe in the Standard Model, they are the partons of the sea = u and d-quarks and the valence partons = c,s,b,t-quarks and gluons, on the other hand, in other theories the proton could be compose by for example of exotic fermions among others new particles), therefore when an electron (or other lepton, or fermion or in general some particle) interacts with the proton, it interacts with some parton inside the proton and this interaction can be explained by the QFT.  
