You can store items and cash in your house. Both the inventory and wardrobe waypoints look the same.
You can move your storage waypoint around in your house. Choose "Move Inventory" from the housing menu, and then move your character around to the spot in your house where you want to move the waypoint to and press G to confirm.
You can use same process as above to move your wardrobe waypoint around. Just choose "Move Wardrobe" instead.
You can store cash, inked/marked bills, and inventory items at your house.
The number of inventory slots your house storage will start off with depends on the house itself (usually between 40-80 slots).
You can expand the number of slots by choosing "Upgrade Storage" in your main housing menu.
See "How to Use the Inventory System" for more information about how the inventory works.