Posted 19 October 2016 - 03:23 AM
The first question is...how do you access/get your email messages? Apple Mail? Some other email client? Using a web browser?
Assuming you use Apple Mail, then the likely easiest and best way would be to use Time Machine. This will backup your entire Mac, including email messages. You can then supposedly restore email messages directly within the Apple Mail client using Time Machine.
If you don't want to use Time Machine, then all of your Apple Mail accounts and email are stored in the following location:
[hard drive name]:Users:[user name]:Library:Mail
If you backup that folder (note that you may have to hold down the Option key while using the Go menu to find the Library folder in your Users account in the Finder as Apple hid that folder a number of macOS versions ago), then all of your Apple Mail settings, your accounts, and your email messages (including attachments) will be backed up. Restoring them might be a bit more of pain as you have to deal with permissions and a few other things, but they will be backed up. This manual method is a bit more of a pain than just using the Time Machine option. I believe you can potentially shortcut some of the hassle by using the Import function, but I have never tried it.
Another option is to export messages or even entire mailboxes. For a single message, just highlight it, then drag it to a Finder folder. This will leave a copy of the message in that folder AND in Apple Mail. For mailboxes, supposedly if you right-click on a mailbox in Apple Mail you will get a pop-up menu with the option to export the entire mailbox...for some reason, it is not working on my Mac...and I don't know why at the moment.
If you are using some other email client, then I still might be able to help. Same goes if you are access email by way of a web browser. I, however, would need to know the specifics.