Bank[]
Bank boxes are local except for pledge rewards and some special items. So make sure you choose the bank spot wisely. You can increase your bank space by either paying ingame gold (100 gold for each 10 slots up to 200, then 200 gold, then 500 gold etc) or $5 on the website for 100 additional slots. Stacks count as one slot but resources come in different qualities and can then take up several slots.
Container[]
Personal inventory does not have an item limit so using a backpack inside your inventory does not increase the number of items one can carry, it only works as a sub-container for example for sorting. Placing a backpack in your bank will not increase the limited total number of available slot but instead takes up a slot of its own. Placing a container in your home allows you to store items there. A container like a chest or backpack or such have weight and take up one slot in a home in the special category of "container indoors" or "container outdoors". How many of the latter you can have depends on the lot type.