In Days Gone inventory management is an important step to stay alive. Crafting Materials are important to make items such as health bandages and Scrap is vitally important to repair the bike.
How to Increase Inventory Limit: Buy the Survival Skill “Carry That Weight” to carry twice as many crafting materials, traps and throwables. Buy the Ranged Combat Skill “Up The Ante” to increase the ammo capacity for all weapons.
These are the only ways to increase the inventory and ammo limits in Days Gone. There are no belt upgrades or anything like that, just these two skills. They are high-tier skills which means you must put enough points in their skill trees (Survival & Ranged) to make these particular skills available for purchase.