For a gas made up of electrons, quantum effects become important. Roughly speaking, it is strictly forbidden for two electrons to be present at the same location (this is called the Pauli exclusion principle), and if anyone attempts to concentrate electrons in a small volume of space, they will start to flit back and forth madly (cf. Heisenberg’s uncertainty principle). Just like with regular gases, this flitting back and forth leads to pressure, in this case to what is called degeneracy pressure.
It is this kind of electron degeneracy pressure that, for instance, stabilizes a white dwarf, preventing further collapse.