Playing Fallout 1 with unarmed

It depends as what you class "Unarmed" as. You can use the Powerfist as an unarmed weapon but then in a literal view its not unarmed. Essentially, it is very possible as I myself have done it and know of others that have to.

There is also a perk you can at around Level 18 which makes all unarmed attacks critical. However, before getting the powerfist, it really is such a grind to be able to deal with things, but its exactly that, a challenge.

Once you have these two though, the game is very, very completable.

If you want to know if it is possible to complete the game with a non-combat oriented character, the answer is yes. In fact you can complete it without killing any characters in combat. You might not gain a ton of experience however, and it might not be very much fun.

If you want to know if unarmed is a viable combat option, I defer to this answer.