(Too long for a comment)
I think the title and the description is a bit vague.
You may want to think about these: Do you want the files encrypted just on the disk or when they're transmitted over the internet as well? What kind of interface can you accept for accessing the files (web page or mounted drive)? Do you want to be able to view/play media files directly or just simple file downloads and uploads? What kind of operation latency can you accept? Do you have enough bandwidth to host this yourself? If you don't can't find an existing solution, do you want to go through the trouble of building, securing and maintaining your own system?
Also, you should consider cloud storage. It may not sound very secure, but as long as you choose a technically capable provider and use a strong password, it's probably more secure than what you can do yourself.
Also, security is not just about encrypting data. Have you considered the cost of losing those data? I.e. would it cost you more if those data are stolen (assuming there're cyber criminals interested in your data) or if those data become unreadable due to the loss of the ecryption key?
In response to the comments:
I think in this case, many solutions are available to you. I suggest you try the encrypted home folder feature that is available with some Linux distributions. This has the advantage over full disk encryption in that the files can only be decrypted when you're logged in (because the encryption key needs to be decrypted using your login password). This does create some additional difficulties when you're mounting the drives but can be easily remedied by running a normal log-in session over SSH.
If you don't like encrypted home folders, there are also fill disk/partition encryption softare for each platform like BitLocker for Windows, FileVault for Mac and various encrypted loop mounts for *nix and cross-platform tools like TrueCrypt. In this case, you must ensure nobody else can gain admin access to the system as they will be able to read all files.
Mounting drives over the Internet is always a problem though. Opening up file access services to the world is a big security risk. Most people would put the file servers behind a firewall and use VPN or SSH to access it. You can hit two birds with one stone if you use SSH.
Protocol-wise, SMB and web-dav mounting are supported on all major OS platforms (Windows, Mac, *nix). However, the former is very difficult to setup over SSH and the latter does not perform very well for large amount of file access, so there'a trade off. Other protocols like NFS and AFP are limited to their own platforms.