The wording used in the comic prevents this from actually being a paradox. The question should require that the barber "shaves all those, and those only, who do not shave themselves."
Removal of the "all" leads to the following logically:
The barber must shave himself, but he cannot, as he only shaves those who do no shave themselves. This is a contradiction and cannot be true.
The barber simply does not shave himself. There is nothing in the statement as-presented which prevents him from being shaved by someone else. This could be true, so we do not have a paradox.