You could connect their email to Thunderbird and export a backup from there. A roundabout way to do it. Who's the email provider? Gmail?
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As others have said you can export the emails to a file. They won't individually be openable though. Also major thing DON'T DEPEND ON A FLASH DRIVE. If it's just a backup fine, but please don't use it as the only place the data is. They die randomly all the time.
What email service do they use? You’re probably looking for an IMAP downloader, unless the service itself offers another way to export data (e.g. google takeout for gmail).