There are still barbers like in the picture above, especially in the smaller localities. Alternatively, the cities have plenty of modern and well-trained hairdressers. The top hairdressing salons will charge about GDR 3000 - 3500 for a men's haircut and between GDR 4000 and 5000 for women, which is a fair bit cheaper than in a lot of countries. More fancy stuff like highlights and perms is not cheap but will be well done.
Appointments are generally necessary.
Avoid Friday and Saturday which are often pretty busy.