Here’s a great feature by Google – they allow you to backup (download) your Gmail, Calendar, Contacts, the contents of your Google Drive, Google Books, Google Voice, Hangouts and other services that may host your content.
This is great if you want a local backup of all your data.
Unfortunately, it seems you cannot ask for a backup of your GMail from a certain date (assuming you’ve already done a full backup of your GMail recently), so you’d have to run this master/full backup over and over again if you wanted to obtain periodic backups of your new emails since your last backup/download.
The process kicks off in the background on Google’s servers and depending on how much data you have to backup the process can take hours or days. Eventually you’ll get an email with a link letting you know the zipping (or tar/gZip ) process is complete and you can then download the files. Google will automatically create multiple zip files if your content is >2GB.
The archive files you create from this process will stay on Google’s servers for one week.
Here’s the link to backup any service: https://www.google.com/settings/takeout/custom
If you want to specifically see a shorter list, you can try this URL: https://www.google.com/settings/takeout/custom/gmail,calendar