I believe that the standard solution is to build a small, low power computer and fill it with hard drives since a good NAS costs a ridiculous amount. There are some very good operating system options, so you are not bound to windows unless you wish to be.
With regards logging into it, it depends rather on what operating system you use, and your technical skills/available time to learn new tricks. VPN is a virtual private network, it convinced your current computer that its directly wired to the physical network (with your NAS on it). This works well, I've had success with hamachi. FreeNX is what I would recommend to actually log into the machine.
If comfortable with linux, ssh'ing into a network machine is straightforward. That only gives you a shell, but sshfs will mount a remote drive as if it's internal, and freenx will allow you to actually sign into a computer on the home network.
FTP/SCP will let you copy files, VPN/sshfs will let you browse it normally. You log into one of the machines attached to the NAS, not into the machine itself. Unless you've gone down the build-your-own approach, in which case you can log into it directly.
Depending on what you have in mind, you may discover bandwidth to be an issue. Streaming video over a mobile broadband connection (for example) doesn't really work, and remote desktop can be very slow.
I'm quite interested in things like this, and happy to help, but must ask for more information on what you wish you achieve. And none of the solutions are that easy as such
