First your closet needs vents or louvers. These are openings into the closet that allow air to enter. The air handler cannot operate in a vacuum. Before you start worrying about floor space, make sure there are vents that allow air to move in. Making sure air can move out is the next step. Closet doors are generally the easiest surface to vent, because they are thin compared with the walls and they allow for simply air supply.
Floor openings serve only one purpose in your system: allowing air to pass from the handler into the ductwork. Your system will decide if you need one of these openings or not. If the ductwork in your house runs under the floors, a false bottom in the closet with an opening for the air handler is necessary. But if the air runs through the ceiling (based on the location of the closet) you will need an opening in the ceiling.
Sizing is important when creating the necessary floor or ceiling opening. The size necessary will depend on your model and how much air the handler is designed to move. For smaller models, 320 square inches may be necessary for a open area into the ductwork. For midsize models, 360 square inches may be needed. The largest models need 450 square inches in order to work properly.
Installing an air handler in a closet comes with a number of other considerations. Generally the unit needs enough space around it to work properly. For furnaces, this means not using the closet for any storage purposes and keeping insulation away from the unit. In addition to clearance, you will also need easy access to air filters or control panels for future work on the system. Your installer or manufacturer should be able to offer further advice on placement.