If something is not allowed on the premises of a school or workplace, then it's not allowed anywhere on that property, whether indoors or outdoors.
All the property at a particular site, including buildings and land, are its premises. At a museum, someone might ask if there's an ATM, or cash machine, on the premises, so they can withdraw some money to give to their kid to buy something at the gift shop. Supermarkets often have signs saying that "all shopping carts must remain on the premises." In other words, don't try to take one to another store or to your home! On premises is a less common form of the phrase, used especially in writing.