We are gradually becoming more and more accustomed to the fact that our data is stored on the servers of others. Barring privacy issues, that's fine too, but what if you'd like a backup copy?
You may not often think about it, but the moment you decide that you want a backup copy of all your mails in Gmail, it suddenly becomes clear that there is no button for that in Gmail at all. At least, it is indirectly there, but it is well hidden. Of course you can always make a backup by forwarding all your e-mail to a Pop account that you have configured in the e-mail program on your PC (where you can set that mails from the past should also be forwarded). However, you can also opt for a one-time backup (which you then perform once in a while), using Google Takeout.
You do this by surfing to www.google.com/settings/takeout and then indicating in the overview which parts of your Google account you want to include in your backup. That can be mail, but also photo albums, blog posts from Blogger and you name it.
Turn off the switches for the items you don't want to include in your backup and click Next at the bottom. Then you indicate how the archive should be saved (.zip is the most useful) and whether the file should be added to your Google Drive or whether a download link should be sent to you. Archiving can take a while, but after that you have your mail (and other Google products) safely locally.