Closely related to Can I edit bank/backpack space with Gibbed's save editor? I really need to store more items across less characters. Is there any way in-game to store piles of loot on a single character without save moding?
Alas, this bug has been fixed, as of a patch right around the Kreig update. If you attempt this you can no longer overstuff your backpack, though if you have done this in the past your "pack mule" will still have excessive amounts of items in their backpack. You can use or remove these items, but you cannot add more backpack items through any normal means.
I accidentally found this trick out; if you withdraw items from Claptrap's secret stash (the place you can trade items between characters) it will allow you to overfill your backpack, like so:
This allows you to store all your items on one spare character like a gigantic bank. I haven't seen an upward limit yet, but it's not less than 90 items. You can do this trick with the Bank too, but the useful thing about Clap Trap's stash is that you can send overflow items from other characters to your pack mule and never worry about storage space.
Your overflowing backpack still functions normally, except you can't add more items to it by buying/picking up items. Quest rewards and Claptrap's stash items will still be added to it, and they can be removed normally. Note this means this is best done to an alternate character being used exclusively for storage, since being unable to pick up/buy guns is a lethal handicap in normal play.
Spend Eridium at the black market in sanctuary on the backpack upgrade. You can also purchase larger capacities for all the various types of ammo.