What height concrete gate posts do I need for a 4 foot high gate?
The post should be 4 feet high above ground and deep enough to be set into concrete below the frost line.  A 4' hole (8' post) should be more than sufficient.  Probably overkill, in fact, but wouldn't go any less than 6' total, 2' below grade, and if you're in the north, definitely go deeper.