The challenge is, "The Throw Away Challenge".
Basically, how it works is this.
Since you start out unarmed you must remain unarmed. This however does not mean you can not use a weapon. You may use any weapon you find but can not pick-up, create, store any ammunition. You may also not repair any weapon.
So for instance, there is a 10mm SMG in Doc Mitchells house. You can pick it up, equip it and use it. Provided it has ammunition already. Once the ammunition is spent, you must throw the weapon away. If you pick up a weapon and it has no ammo obviously you can not use it. Same applies to broken weapons. You can pick-up and carry any weapon but once you equip it to use once you are done using it you must throw it out.
This rule applies to "unique" weapons as well. Once the ammunition for them the weapon must be tossed. It also applies to melee weapons. Once a weapon breaks it's garbage.
Melee weapons can only be used during that combat situation then tossed. Again, you find a shovel and are immediately attacked. You can use the shovel to defend yourself but must throw it away. On the flip side, a shovel is also a tool so if you know you are going to use it to rob graves it can only used to rob graves and you can not use it as a weapon. Once you use it in an offensive or defensive position it becomes a weapon and must be thrown away. Your fists are your default weapon since they can't be thrown away.
I came up with this while watching a Youtube video of a fan film for something. The characters were picking weapons up, using them and throwing them out as soon as their use was over. I figured it'd be a challenge in FO.