He goes to a barber downtown. (Merriam-Webster’s)
Does a barber mean a person or a shop?
It could refer to either a person or a shop, but probably a person. However, nearly all barbers work in barber shops. So, in effect, the person is going to go to a barber (person) who works in a barber shop (place) that is located downtown.
Well the dictionary indicates a barber is a person, not a shop. Should I infer that you doubt the definition in dictionary? The term for the place is "barber shop" or "barber's shop". But yes, the implicit meaning is that he goes to a barber shop; this is accentuated by the lack of otherwise specification and by the presence of the article "a". In the same way, when you say "I go to the dentist" usually means I go to the dentist's office.