This is mainly focused on websites, but you can launch other items that don't require any ports exposed:
http://www.octohost.io/data-stores.html
I added some "magic" Dockerfile comments - this is the one to add in order to not expose anything via HTTP:
NO_HTTP_PROXY
I am not a huge fan of volumes where data is stored on the box - but you can do it the same way I've described on the page.
I think of octohost boxes as app servers - if you have data on it, you should likely have it stored elsewhere. I've used Heroku to store additional data sometimes:
http://blog.froese.org/2014/03/17/using-octohost-with-heroku...
We've also used remote MySQL servers:
https://github.com/octohost/mysql-plugin
There's lots of ways to do it.