Google Takeout – Where is the Download Button?

You could try out HTTrack

Full description from their website:

HTTrack is a free (GPL, libre/free software) and easy-to-use offline browser utility.

It allows you to download a World Wide Web site from the Internet to a local directory, building recursively all directories, getting HTML, images, and other files from the server to your computer. HTTrack arranges the original site's relative link-structure. Simply open a page of the "mirrored" website in your browser, and you can browse the site from link to link, as if you were viewing it online. HTTrack can also update an existing mirrored site, and resume interrupted downloads. HTTrack is fully configurable, and has an integrated help system.

It is available as WinHTTrack for Windows 2000/XP/Vista/7 and available as WebHTTrack for Linux/Unix/BSD

If you are up to the challenge you could try forwarding each and every message to a new email address, but this would require some tricky filtering as well as header rewriting on the backend.

Here's what I would suggest:

1. Create a filter in Gmail. In the Has these words: textbox type "is:read". The action should be to forward to your new email address (which must first be verified). You'll notice Gmail warns you that searches with is:, label:, or other related operatores are not applied to incoming mail, but you will be fine. You might want to create the corresponding search for new email first where you could wildcard the To/From.

2. At this point you would have all your emails forwarded one by one to the new email address, along with future incoming email also forwarded. The tough part is that its a forward and not a mail redirect which would be much nicer, keeping headers intact, etc. At this point you'd have to use some text processing or third party programs to perhaps attempt to get rid of the Forward fluff and potentially rewrite headers if it interested you.