Why would you want to move your files to the Internet? Backup? Safety? Sharing? Anywhere access? No matter which one you choose, storing files on the Internet, or cloud storage if you will, will give you all the comfort that I have counted above.
Although the cloud services are relatively new in people's time, but in the “teenage" period of Internet time, we have seen many companies offering many features, but then closing all their services. One of them is AOL-backed X-Drive, which stopped all its services in January, 2009.
You might see other services on the Internet offering online storage solutions. Some of them are still not mature enough, such as Microsoft Live Space (you can't create subfolders and there is no file history), and others of them still stand. I have chosen the ones that are still standing in online storage area (as of today, of course), but frankly, I have left out Microsoft Live Space for now.