What options are there for permanently connecting a rooted Android Phone to a NAS?

I keep all my documents, pics, music, etc. on a home NAS so I can access the data from around my home. For work I mostly use DropBox. But as my free 50GB are going to expire soon, I need to find another solution. So I thought I could move all my stuff from DB to my NAS and access it from there. I could use VPN but they aren't allowing outgoing connections and won't enable it for only that reason, which I can fully understand. BTW I have a FritzBox 7390 as router and I plan of purchasing a low-power Shuttle PC for simple server-stuff. My Nexus 5 is rooted and runs the latest OmniRom build. Are there any options here? If possible, it would be nice if my NAS could be mounted so I can move files around easily with an app.