In order to store weapons in Days Gone, you first must purchase a weapon from a merchant. Once bought, you will then be able to store it inside a weapon locker at an encampment.
All weapons you buy will also be available via the weapon locker at encampments - this also includes any Pre-order bonuses and DLC weapons.
You also can't store any of the weapons you can find lying around in the game. This is really unfortunate as you often find better ones than the ones being sold.
So if you want to store or keep these weapons, you will need to keep them on you, as if you drop them, they will be gone forever.
However, you can unlock better ones by doing various missions and earning trust with survivor camps. You can also gain trust by doing the following.
- Hunt down freakers and turn in their ears at the Bounties station
- Give items to the kitchen
- Complete Main Story Missions
- Complete Bounties, Campers in Need, and Supply Runs
When you get high enough trust, you will be able to get better guns, among a ton of other items and upgrades for both your weapons and motorcycle.