From Wikipedia:

The barber is the "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself?[1]

Any answer to this question results in a contradiction:

The barber cannot shave himself, as he only shaves those who do not shave themselves. Thus, if he shaves himself he ceases to be the barber specified.

Conversely, if the barber does not shave himself, then he fits into the group of people who the specified barber would shave, and thus, as that barber, he must shave himself.