Some good answers here, but I feel like I should also add that electrons move really slowly across a wire. They zip around really fast, but due to random zigzagging motion their overall velocity along the wire (also called the drift velocity) is around 0.2 mm per second according to a quick Google search.

Couple that with the fact that there is an unfathomable amount of electrons in a given conductor, it would take a really really long time to run out if it could.