Two ways to save your files:
1) Download individual files
2) Use Google Takeout
Google Takeout is great, but the CATCH (referenced in the video) is that it puts all your email into one .mbox file, which can be challenging to get the emails imported back into your account. If you only have a few emails you want to keep, it's recommended that you simply FORWARD those to another email address that you own before June 3, 2015.
However, if you have lots of mail you want to keep, here is a link to a tutorial on how to import that email back into another google account at a later time.
https://backupify.zendesk.com/hc/en-u...