Having a backpack limit makes sense for combat, but perhaps let the player digitally store it on an in-game computer (accessible in Sanctuary).
why would you? a maxed out inventory and bank provides me with enough storing space for my gear. and if not i mostly store stuff on a low level character
That would be a lot of data taken up with how many weapons are in the Borderlands franchise, and that is not counting the different variations that each weapon has. Besides, there is already something similar to what you are proposing in the form of the giant safe that is in Crimson Raider HQ.
Personally if I were to modify anything it would be Claptrap’s locker as it only holds four items at a time. If and when you are moving gear from character to another it becomes quite tedious, especially if you are exchanging equipment.