The Story of the Barber

In a certain remote village, there is a barber who shaves all and only those villagers who do not shave themselves.
The story is simple, but now comes the question, "Who shaves the barber?"
If the barber shaves himself, then he does not (since the barber shaves only those who do not shave themselves). If the barber does not shave himself, then he indeed does (since the barber shaves all those who do not shave themselves). So, the logic shows the barber shaves himself if and only if he does not shave himself, which is totally absurd.

The only possible answer to the question is that there cannot possibly be such a barber, nor such a village. Put simply, the story is impossible!