With lots of new features being added to web emails, there is definitely stiff competition, but none of the service allows you to back up your emails. Backup option would be very useful when you shift from one web email to another. Mail Carbon is a simple free tool written in Java which allows you to back up emails from one server to another using IMAP. For example, if you are shifting from one Gmail account to another one, then mail carbon can get all your emails to new account.
You need to provide the source and destination hostnames and user credentials. If you are using Gmail account, you can check the option which will select the protocol and port automatically.
This simple and yet useful tool has both command line as well as GUI. In case you are using it in Vista, make sure you run with admin privileges. In case of mails are archived using folders or labels (as in case of Gmail), this tool will create these folders on the back up server as well.
We tried backing up our emails from Gmail account to Google apps account and it worked fine without any issues.