It is entirely possible to make razors from flint, crystals and other natural occurring materials. Flint blades can be extremely sharp, especially by skilled craftsmen, which they obviously were at the time when that was the common material.

Can I add to OPs question, about hair cutting in general too?

Follow-up question: what about cutting nails? I can see how one could relatively easy cut their hair and beard, but nails on the other hand.... One would've need to have some kind of scissors, I'd think.