Have a look at DriveHQ.com which offers free and paid-for services.
Whether a service is right for you depends on how much you want to transfer (and whether your ISP has limits or adds charges) and how knowledgeable you are about content on your PC.
For example, if you have MS Office, and a whole stack of bought software, there is usually no point in copying it onto backup storage if you expect to download it onto a new computer as the installation procedures build information into a Windows database (the Registry) and without doing that properly, the software won't run reliably (or at all, more likely).
Copying data in the form of documents, downloaded items (eg software installation programs), and so on, is fine. Copying any music or video that you have 'ripped' from CDs/DVDs is in a different league as that may be against the terms of the online service and would also take a lot of time/space.
If you do have items which you cannot replace then making them into a file archive (eg using ZipGenius) may avoid them being spotted as MP3 files for example, but they will still be large and take a long time to upload.
For anything over say 5 GB you will probably have to find a paid-for service. I have a few suggestions but not to hand.
How much data do you expect you would need to store ? I am assuming family photos etc and letters/ work related docs rather than copies of DVD-held films....