Chests are placed and picked up much like crafting surfaces, and do not interfere with a Blockhead's passage.
When a Blockhead selects a chest and "Use Chest" from the pop-up menu, it opens a storage space of sixteen storage slots in a four-by-four square, each able to hold a stack of one type of item. Different stacks may be of the same or different items.
Chests can be placed inside baskets or other containers. They continue to carry their contents, and can be stacked with other chests even if they are empty or have differing contents. When a chest is later placed and used, its contents at the time of removal are available again. There is no way of identifying the contents of a chest before using it, so there is no guarantee that placing a chest from a stack will get you the chest you want, although general inventory stack handling tends to be LIFO (Last In First Out).