I'd say that most barber chairs would allow for transfer of someone requiring accommodation. Once they transfer to the chair, the chair can be raised/lowered to allow for the use of the hair washing sink. So, from a patron use, I do not see a requirement much more than the typical accessible route and restroom.
From an employee standpoint, I'd say that the stations are an employee work area, thus 1103.2.2 applies (below). If a stylist/barber were hired, and they required a lower sink, counter, or other accommodation, then they can request that the employer provide pursuant to the ADA as a reasonable request for accomodation.
1103.2.2 Employee Work Areas
Spaces and elements within employee work areas shall only be required to comply with Sections 907.5.2.3.1, 1009 and 1104.3.1 and shall be designed and constructed so that individuals with disabilities can approach, enter and exit the work area. Work areas, or portions of work areas, other than raised courtroom stations in accordance with Section 1108.4.1.4, that are less than 300 square feet (30 m2) in area and located 7 inches (178 mm) or more above or below the ground or finished floor where the change in elevation is essential to the function of the space shall be exempt from all requirements.