Is there a point of balance where the gravitational pull of a sphere of electrons is equal to their electromagnetic repulsion?
That is to say, could it be possible to create stars that are made purely with electrons and that are stable and don't fly apart?
No, it's not possible:
The electric force between two electrons is is $2.4* 10^{43}$ times (trillion-trillion-trillion-trillion-trillion times!) stronger than the gravitational force between these two electrons. So if you add more and more electrons to the "electron star", the gravitational pull will indeed increase, but the electric repulsion between the electrons will also grow, and remain $2.4* 10^{43}$ stronger than the gravitational pull.
So a realistic star will need to be almost exactly neutral to have any chance of remaining intact. For example, the sun has around $10^{57}$ electrons and also $10^{57}$ protons and additional fewer neutrons (which, as their name suggests, are neutral).
