Using Desktop Sharing Software
There are free desktop sharing software like TeamViewer and UltraVNC that also let you remotely access your computer files from anywhere else. LogMeIn Pro and CrossLoop Home are some other good options for transferring files from a remote computer but these are paid services.
If all your computers are running Windows, you can use the built-in remote desktop connection feature of Windows (available in XP and later) to access your home computer from another Window computer over the Internet.
With Desktop Sharing software, you’ll always have access to all your files but one big limitation is that it requires that the remote computer stays on. Also, none of these options will help you access files on a mobile phone except LogMeIn which has an iPhone app.
Using Online Backup
You can use an online backup service like SugarSync, Carbonite, SkyDrive or even Dropbox – they provide desktop utilities that will automatically upload files on your home computer to the Internet and you can then access these files through the web browser of any other Internet connected computer.
The upside is that your files will always be available to you even if the home computer is not running. SugarSync and Carbonite have a mobile optimized website so you can access files from any mobile phone as well.
The downside is that if you are using any of these services for the first time, you’ll to wait until they upload all your data to the cloud and this process may take long if you have too many files on the home computer.