This is probably the most common approach and should take just a few minutes to set up. Gmail offers access to all your mail through desktop clients such as Apple Mail, Microsoft Outlook, Mozilla Thunderbird, and many others. The latter is free and available on all major operating systems, so if you're doing this for the first time and especially if you're only planning to use it for backups rather than as your primary client, then Thunderbird might be the way to go.
For those unfamiliar with the setup process: After signing into Gmail, click the "gear" in the upper-right corner, choose Mail settings, then load the "Forwarding and POP/IMAP" tab. Enable POP for all mail and set Gmail to keep its copy.
You'll then need to setup your desktop mail client to retrieve your emails. Google offers step-by-step instructions for a number of clients and Thunderbird's automatic configuration is usually spot on getting your settings right with just your username and password -- but make sure to use POP instead of IMAP for one-way message transfers.
Messages will be downloaded automatically each time you launch Thunderbird and stored into your profile folder -- which you can back up standalone as well for some added security. It's pretty straightforward, but for those who prefer to use the web-based Gmail interface, remembering to run Thunderbird or whatever desktop email client every once in a while might be more work than they're willing to deal with.