The Bank Box is a place of storage that every character has access to from the moment they are created. It is the only means of storage which is considered 100% safe.
To open your bank box you must first visit a bank. Most cities have one, usually found in a central location. Once there you need to find a banker. A common mistake of new players is thinking that the bank itself determines whether you can open your box. It is actually the banker or minter you must be standing in range of.
When you are close enough to a banker or minter NPC, say "bank" to open your box. You can also open it by using a context menu on the NPC. The context menu method won't reveal you if you are already hidden.
The bank box has a limit of 125 on items (150 if you have purchased the account storage expansion upgrade or 175 with the high seas booster and the account storage expansion upgrade). Every item placed inside counts towards this limit including containers, gold and checks.
If you are short on space combining multiple piles of gold into one check can help free some up.
At one point in early UO history bank boxes also had a weight limit of 400. Gold did not count towards this limit.
Over the years Houses went from a completely unreliable method of storage to being almost as secure as the bank is. If you don't allow anyone access to your house and keep everything in owner secure containers it basically is. However the contents of your bank will stay with you indefinitely when your account is deactivated, a house will decay in 90 days.