A simple external harddrive with a backup utility would be able to have anything and everything on your computer saved on it.
There are many ways to do it locally, on LAN or on internet. On internet there are plenty of free trusted sites available that store your files and permit you to access them from anywhere. the sites like google, yahoo, msn are leading in this. there are certain paid sites that provide to storage space on internet for the same purpose. On LAN, you can dedicate a single PC for all users to store their simple/complex files by creating users folders and giving them access accordingly. On a single PC as said above you can add a dedicated large sized harddisk dedicated for the purpose. But take care that users have access limited to their files/folders only, otherwise there could be lot of issues like data theft, data deletion etc.