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.
First, you need a "DynDNS" account, so from outside, you can connect to your home network without knowing the (daily changing) public ip.
Then, you install ssh on the NAS (I guess you will use linux, e.g. ubuntu server lts).
I'd recommend forwarding another port to ssh, like 12345, in order to stop at least basic, automated scans from finding your ssh server.  

As client, I would recommend ES File Explorer.
Connect to your NAS by sftp.  It's using strong encryption.