You can run a Postgres on Fly.io full time for just slight more than $2.

It's not a great business though. The compromise you're making is that we don't manage it. It'll probably stay running, it's likely to be worry free (especially if you barely use it), but if anything goes wrong it's up to you.

You could also spin up a VM for cheap somewhere, install Postgres, and just let it be.

Managed databases are expensive to run – worry free costs quite a bit of money to ensure. Maybe you don't need that for a hobby project?