Software like Mailstore ( https://www.mailstore.com/ ) can be used to download email. If you have your email client configured to use POP3 instead of IMAP, it will have already downloaded your email. There are many other 3rd party software options that can also be used to download email as well.
While we do not have general instructions on downloading email, we do offer information on how to transfer to a new email host. We are also able to provide a backup of your cPanel account which will contain all your email in a large zip file. This will be in a standard cPanel backup format easily handled by other hosts.
If you decide not to download your old emails, but you do get a new email host, many hosts will help migrate email to their service. We can provide an email transferring service to handle this for you (please keep in mind it will be billed at the standard hourly rate). If you don’t get a new email host, don’t download them, don’t do anything, we can at the bare minimum provide you with a download of the cPanel backup which contains all your emails. Otherwise, the emails would be lost.