The short answer is that no, you won't be able to access the hard drive from the internet. In general, this is a good thing, as you don't want people snooping around your files...so some sort of secure connection is necessary.
If your NAS device has an FTP server (I'm not sure whether any do), you can tunnel FTP connection into your network (preferably over SSH, as this provides encryption which prevents your data from being read as it travels over the Internet). If you can't get a NAS with FTP, you can turn an existing machine into a NAS by putting a hard drive in it and setting up the appropriate server software. If neither of these is possible, then you can get a NAS and do something more radical, like tunneling Windows file sharing connections over SSH. I doubt your Belkin router has a VPN server, but that's probably the best option of all these if it's available (it doesn't require an SSH server).
I have to warn you that none of these options are easy, mainly because of the need to set up an SSH server on your home network. If you want to try any of these options, I'll be glad to help you with them (and I'm sure others will too). However, my time is limited, and ultimately the burden rests on you to understand what's going on. A misconfiguration could cause the project to fail in unexplained ways, or worse, expose your NAS to the internet so that some idiot can break in and steal all your files.
Just a heads up--I'll be away from my computer for the next two or three days.
Cheers
EDIT: FreeNAS is a good choice. I would still tunnel the connection over SSH, as you don't want to transfer files over the Internet unencrypted. VNC is not an acceptable choice because it provides no file transfer capability.