The trunk is a large chest located in the living area of the Observatory, between the desk at which you can save and the door which leads into the Courtyard. You are able to store Items (apart from Flesh) into the trunk. Unlike the bag, the trunk does not have a limit on how much can be placed inside it, so if your bag is full it is best to come back to the Observatory and transfer Materials and unequipped Equipment into the trunk, as well as anything else not required in the Towers such as gifts.
When giving Elena a gift or selling items to Mavda, you are able to select items in the trunk without moving them to your bag. However, the rest of the time you are required to move items in and out of the trunk by interacting with it, and you are unable to even view the items.