The below instructions will help you make a back-up of the contents of your Mac Mail mailbox.
1. Open the mailbox you wish to back-up in Mail.
2. Click on Mailbox
, and then Export Mailbox
2. Choose a folder or create a new folder where you want to store the exported mailbox, and then click Choose
Mac Mail exports the back-up file as a .mbox extension. If you have exported a mailbox before, Mail will not overwrite the existing .mbox file, but adds a number to the filename, to create a new version. For example, "My Mailbox 3.mbox."