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The barber could be a woman, a prebeard child, an American Indian, skin can't grow hair due to fire accident on the face making it impossible, etc. Essencially, any situation where a barber growing a beard doesn't apply.
Let's assume the barber can grow a beard (some women included), and shaving includes all methods of removing hair (including things like fire to the face), then the solution is still obvious. One of premises in the original statement is false: "Do you shave yourself? If not, come in and I'll shave you! I shave [everyone] who does not shave himself, and noone else."
One of these premises are false:
1. Everyone in town has a clean shave, which includes the barbor. (so the barbor has a beard)
2. "I shave [everyone] who does not shave himself..." (so he doesn't shave everyone who does not shave himself)
3. "...and noone else." (so he shaves others too)
Bottom line: the orignal statement has to be false.