If you strip the electrons off some atoms and use just the electrons[0] as reaction mass, you will eventually get a large enough electric charge you can no longer throw the electrons away from you. Electric forces behave similarity to gravity, so while it wouldn’t normally be phrased like this, you could say your engine exhaust will eventually no longer have escape velocity from your ship.
For this reason, ion drives do things to neutralise the net charge.
(If you meant using them as a power source rather than reaction mass, it’s technically possible but that’s called a capacitor and they have very low energy density).
You didn't address this part of the parent comment:
> I imagine it'd pull in electrons from all around itself, but I don't know how the numbers come out.
I never thought of it before but it seems like that should work. "Space" is actually a neutral plasma, right, so it should be full of free electrons. Those should neutralize the ship before any significant charge builds up. It seems like you should be able to use space itself (or more accurately the interplanetary medium) as a massive ground plane to complete the circuit for the charged exhaust beam.
Space is pretty empty. From what I can find, the interplanetary medium is around 5 particles/cm^3 compared to the exhaust from the ion thruster which seems to be around 10^6 particles/cm^3 and disperses to 10^4 particles/cm^3 further away.
"As the charge-exchange plasma
density near spacecraft is at least about three orders of magnitude larger than the solar wind plasma density, the plasma environment of DS1 spacecraft is completely dominated by the charge-exchange plasma in the plume."
For this reason, ion drives do things to neutralise the net charge.
(If you meant using them as a power source rather than reaction mass, it’s technically possible but that’s called a capacitor and they have very low energy density).
[0] or, by symmetry, just the nucleus.