If you see an alert telling you that Backup requires a .Mac membership, it could be caused by a network issue, the files you are backing up, or software you have installed.
Home network
If you have a home network, simplify it by connecting the computer you are using directly to the modem, then try to back up from there. If you are able to do so successfully, some element of your network may be causing the disconnect.
Proxy servers
Contact your ISP to see if it uses proxy servers, which may prevent the Backup application from communicating with the iDisk servers.
Specific files
It is possible that a certain file may be causing the issue.
If the error still occurs, continue to the next troubleshooting section. If the file backs up OK, you have found the cause. It's one of the files you are backing up. Try adding more items that you want to back up, and test again. Keep trying until you isolate the item causing the issue. If you create a new file with that data, it will usually resolve the situation and allow you to back up the data.
Norton Internet Security Software
If you have Norton Internet Security Software installed, see Symantec's article on this topic.
Keychain settings
There may be a conflicting or damaged entry in your Keychain.
If deleting the Keychain entry does not fix the issue, use Keychain First Aid. Keychain First Aid is available from the Window menu in Mac OS X 10.3, and for Mac OS X 10.1 through 10.2.8, you can download it.