Is the Pauli exclusion principle also involved in free electrons? Imagine I want to make a laser of electrons like a laser of light. Is that possible, or does the Pauli exclusion principle prohibit that?
 A: This is almost a duplicate then of Pauli exclusion principle in an electron beam. Almost because it asks about cathode ray beams. The answer there is yes; the Pauli exclusion principle plays a role similar to the neutron star role.
For an accelerator beam, where the electrons and positrons are considered free particles, as were the LEP e+ e- beams, the effect has not been considered as far as I can see. As the other answer states, there is no time constraint in such beams.
A: Lasers operate via the stimulated emission of light. This is a phenomenon that applies to bosons, since the stimulated photon is in the same state as the photon that stimulates it.
The same thing cannot happen in fermions; instead there is an analogous process called stimulated absorption which means that the intensity of a fermion beam would be decreased whilst travelling in the "lasing medium". Thus, no amplification.
If on the other hand you are just asking whether degeneracy imposes a limit on the density of an electron beam where the electrons have a narrow range of momentum, then the answer is yes and is handled by the duplicate.
A: Yes, you can make an electron beam. And the Pauli exclusion principle doesn't prohibit it.
According to the Pauli exclusion principle, two identical fermions (particles with half-integer spin) cannot occupy the same quantum state simultaneously. Here you may have missed the word simultaneously. An electron can have the same position in space (all quantum numbers same), but at different times.
