You can use Outlook or Thunderbird or another desktop email client to download your Gmail emails as POP3, which will actually store the messages locally in your email client. You can keep the messages in the email software or, better yet, copy the important ones to a folder on your hard drive. You'll need to enable POP3 access in your Google account settings, under "Forwarding and POP/IMAP"; you'll also find configuration instructions there for setting up POP for Gmail in your email client.
You can also set up Gmail in your email program as IMAP. This, however, basically syncs your email from the cloud to your computer, so in case all your emails disappear from Google's servers (or another webmail provider), your email client might actually sync to the empty server and delete the local copies. If you do access Gmail via IMAP, you can drag or save the messages locally to your hard drive as a backup. Of course, you'd need to do this regularly--before any problems on the web server happen.
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