You can't self-host it. Yes, you can connect to "private" GitHub repo. Pricing model is one time payment for native apps and free web app.
The FAQ says "To accomplish this scenario select generic Git notebook provider (instead of GitHub) and for the password field put fine-grained personal access token which can be generated to have access only to certain repositories."
I created a PAT with EVERY permission within a selected repo, to the fullest-extent allowed by the fine-grained PAT, but still see "An unhandled error occured, please try again" when setting it up within NotesHub.