Do you need to back up Google Gmail before doing a clean reinstall?
No. GMail is a Web service, meaning that all your email and settings is stored on the Internet (on Google's servers) rather than on your computer. This is why you can access GMail on multiple computers, and from any operating system (Mac, Windows, Linux, and more). So, when you reinstall, you aren't affecting what's on Google's servers, so your emails are safe.