You can check out Dokku, which is the open source, single server Heroku equivalent. You can mount a storage volume to the container and access it from your app running in Docker. Dokku makes it stupid simple, so it might be worth reading the source code to see how they do that.
I use Dokku, but it also has a Postgres plugin which is basically just a few commands. That also gives you the niceties of dokku pg:backup etc. If you’re on Dokku anyway then that’s what I would recommend.
For me, SQLite remains the perfect file format for command line tools.
