with linux you have a million ways to do what you want.
Which way you really prefer comes down to what exactly you are trying to achieve - what level of interaction with the server that is.
If you want to share shares from these disks through the internet, You primarily need a dynamic dns. So read up on that and check out dyndns.com OR no-ip.com (others exist as well). The service is free and what you get is a fully qualified domain name f.e.
myhome.no-ip.org, which resolves to your homes ip address. So you can use this FQDN from anywhere in the world to access your home server.
Afterwards (coming to what you seek),
of you want to have access to your files from your webbrowser, so as to upload/download them (or stream them directly), you need a service like webdav. It is also probably safer to use webdav along with ssh. If I understood correctly you have a Mac so this should help. http://hints.macworld.com/article.php?story=20040920100413622
If you just want to download/upload file you also have the option of ftp (or rather the safer derivative sftp). Enable ssh on your server, do the necessary port forward on your router and get filezilla (or other) client on your far end to connect to your server.
across permanent networks, f.e. say from your friends houses, you can set up a vpn server on your fileserver, and give intranet access to your friends. Then, you'll be able to see your fileserver from your friend's computers as if the former was part of their home networks...
The above are rather universal solutions that can be used from Windows, *nix, linux and Mac OS users. So you find among these options the one that suits you best.
Of course one should not forget the options of cloud storage. The option of storing non crucial files online, where you can access it from high bandwidth locations is really catchy and should not be overlooked.
I am sure there are other ways that I overlooked. But I hope This provides a few possible paths.