What you are talking about (save on computer -> save on remote drive) is called folder syncing (specifically, one way syncing). Now, there's not a good free folder syncing program out there. Chronosync Just Works but you pay; free syncing is best achieved by an rsync script called by chron (google it; lifehacker did a piece). If you rsync to a folder that is served from the internet, for instance, from a directory on your web hosting server, you get the functionality you're talking about.
I'm pretty sure rsync can work with the ftp protocol, but it doesn't use the ftp protocol by default (it uses ssh by default, and chances are you don't have a shell account on your webhost). You'll have to read about rsync.
If you didn't have a web hosting package, I'd say you're out of luck. I don't know of any free drive sites. Cuz, you know, it costs money to make a drive available through an internet connection. How would a company generate revenue from the service? You won't be going to a website to get your files, so you won't be looking at ads.