Hive does not provide hosting or database structure as a service. HAF is a database that serves Hives already existing data from the operations. It was made to make providing Hive APIs more easier.
You could run a HAF instance and store your own data in that database too if you also need the data that is in the HAF database which will probably require coding. Otherwise what you are looking for is not here.
RE: [Update] Public HAF, HafSQL, Hivemind, and Hafbe database