How can I backup emails on server to the cloud?

It depends on exactly what you're looking for. If you want offline storage, that's probably not too difficult to pull off. You can simply copy the mail store (or more likely, some subset of "archive" folders) to the other storage and then delete files from the Linode.

If you want the mail to still be accessible, that's a different story. This case essentially boils down to "how do I mount extra storage on my Linode" which comes up occasionally on the forums. I have no personal experience, but according to posts here, people have used Amazon S3 which can be mounted as a filesystem, though speed and reliability seem to be troublesome. In pretty much any case, accessing off-site storage is going to be rather slow. The typical conclusion of these forum threads seems to be "upgrade to the next size Linode," but it sounds like that may not be viable for you. If having lots of e-mail storage is an expected use case, you might be better off moving your e-mail server elsewhere; you can still run the rest of your services on Linode. (Actually, it looks like Dovecot's proxy feature can be implemented on a per-user basis, so you could just offload the large account onto a different server.)