Answer
No
Work Step by Step
If there was such a barber, who would shave him?
If he shaved himself, he would be one of the people shaven by this legendary barber, who $only$ shave people that do not shave themselves: obviously a contradiction.
If he did not shave by himself, he should be shaven by the barber, since he shaves $all$ the people that do not shave themselves: another contradiction.
Therefore, such a barber cannot exists.
