How is the emergency exit row handled when there are no passengers aboard eligible to sit in an exit row seat?
In such an extreme case, the row would be unoccupied.

Nothing says that the row has to have people in it; I've had flights with under a dozen passengers, and if everybody had seats up front, the row would be empty, and that's okay.

You can't have unqualified people in the Emergency Exit row -- that's the limitation. Nothing requires the FA's the move qualified people into the row. That typically does happen (and as has been pointed out elsewhere, that row is often desirable due to legroom), but it's okay if the row remains empty.