I heartily recommend Dropbox.
they start you of with 2GB free, a little more if you register with an educational domain address (*.edu). And there’s ways you can increase that limit by referring friends , giving them a shoutout on facebook or twitter which would net you about 250MB more space each with occasinal promos like a quest full of puzzles that will get you significantly more permanent free space.
You can live on the web client on the website just fine, but it’s big sell is that it syncronizes backups with any machine you want. The desktop app creates a folder in your “My documents” or Home folder, to add stuff to the cloud storage, you just move files to it or save to it like any other local folder. It’ll upload in the background and you’ll have access to it from anywhere, through the webpage, through desktops/laptops with the software, or even from your iPhone or Android phone. Seems like having multiple backups would work pretty well for your situation. And there is no filesize limit from what I’ve seen. You’re only limited by your space remaining and bandwidth for the month (which isn’t noticeable unless you’re publicly sharing something pretty heavy)