Bank Vault or just The Vault as they say in-game allow you to store spare items in there so you can free up inventory slots. The first slot is free then other you have to buy or if you are lucky you can get one through the monthly calendar.
In the village the vault can be upgraded through a daily project. This gives eventually a 9 x 56 spots to store whatever you want.
The mayor CAN set some spots as private or not open to every houseowner in the village.