Further to the answers given, no one has asked an obvious question.
Which part of the Stone age are you asking about. New Stone age using Flint for cutting hair and I believe shaving were quite possible.
It has been postulated that Hair cutting may well pre date New stone age so going back to old stone age, however, you basically need the technology and we don't know enough about how they used some of their stone tools, and the comb is New stone age being around 8,000 years B.C. the earliest example, which is quite late, but hair grooming will have gone on anyway.
It has always amused me that Stone age peoples are shown with less than long hair, and in fact both men and women would probably have long flowing hair, unless it was curly? And we don't know if that was the case or not.