There are a few ways to do specific things, but I don't think there is a 1-stop service that will just backup/sync all your stuff. Before I get going I'll add a big "AS FAR AS I KNOW" disclaimer onto this advice - I'm very happy to be corrected by others who know better!
You can use dropbox to sync a folder of your home computer with the cloud, and then access it online from the Chromebook. You can edit text files stored this way using SourceKit on the chromebook - this is nice if you do any web development work but is otherwise fairly limited. Dropbox also provide a way to download individual synced files, but the native editing ability is limited.
You can use Picasa, Flickr, Photobox or any of the other online photo sites for your photos. Most of these won't automatically sync your computer - you have to upload each group of photos manually as far as I know.
You can upload documents to Google Docs but again this is a 1-off upload rather than a continuous sync.
Music can be uploaded to the google or amazon services, but they aren't available in the UK so I can't really advise you on them. I use subsonic to serve music from my home computer out to my android phone and chromebook using my home broadband connection. This is a bit more complicated to do, but has the advantage of being free and not limited by storage. It isn't a backup though, obviously.
You can also use a remote desktop solution to simply use your home computer from the chromebook. See my other thread for a summary of where this tech is at, but I've found remotespark to be the best free choice at the moment.
I also thought it would be a good idea to use a cloud storage for the most important data... which I don't want to loose! I also found a useful site with the help for cloud back up. Maybe it's helpful for you too?
I think I will try out Dropbox! So many people are using it already, somehow it has my sympathies