I wonder that because during a dialogue in Django Unchained Leonardo Dicaprio speaks about it. However I know that the job of barber was kind of important in Europe for instance so it seems quite unlikely. So I would like to know if it was the case in the deep south or if it's only an element of the scenario.
Sorry if there are any grammar mistake, it's quite late.
I encourage you to read the book Knights of the Razor by Douglas W. Bristol. In the book, he explores this very subject.
To answer you, yes there were slave barbers in the antebellum south. In fact, there are multiple testimonies of white men going to slave barbers to have their chin's shaved. This act, of allowing a slave to hold a straight razor right against their neck, can be read either as an act of bravado in taking such a risk; or as an indication of self-assuredness that a slave would never have the temeriy to cut a white man's throat.
However, black barbers were not only slaves. Many barbers that started out as slaves were able to capitalize on this intimate association with the white elite to gain their freedom and establish barbershops as freedmen. Indeed, barbering was one of the few employment opportunities open to freedmen, and coupled with social norms that considered servile service occupations like barbering to be "demeaning", black barbershops often competed successfully against white barbershops in southern cities like Charleston, St Louis, and Baltimore, as well as northern cities like Boston and Philadelphia.