What would happen if I put two identical particles close enough? Is there a repulsive force between these two particles to prevent them from being in the same point? I mean, in order to obey Pauli exclusion principle?
 A: No, there is no repulsive force in the way that you would describe coulomb repulsion as a force. It is sometimes interpreted as an effective force, but it isn't and there is no exchange of virtual particles as there are in fundamental forces.
The PEP does not prevent ideal, pointlike fermions from being squeezed arbitrarily close together. Indeed, they can occupy the same position if they have different spin. What it prevents is those fermions occupying the same quantum state or for more than two (spin-up/down for half integer spin) to occupy the same location in phase space (momentum $\times$ position)$^3$.
To get fermions very close together means they must occupy different spin/momentum states. It is this spread of momentum that is responsible for degeneracy pressure in a dense, degenerate fermion gas.
A: The effect you drafted exists in reality. 
For a gas of electrons, effects of quantum theory are noticeable. In simple terms, it is forbidden to have two electrons (or other particles) in the same place - as you mentioned. Any attempt to bunch electrons together in a small volume  causes a pressure, the degenerating pressure. If you take only two particles, you get a force instead.
This degeneracy pressure of the electron gas  prevents a white dwarf star from collapsing further.
