2 factor verification on computers too old, i.e. 2008 apple imac

Please determine your iMac's macOS version: Find out which macOS your Mac is using - Apple Support

I can confirm 2FA works just fine with El Capitan, which a 2008 model year iMac ought to be able to run. To upgrade your Mac to El Capitan, read How to download macOS - Apple Support, and scroll down to El Capitan 10.11.

Apple says it will not work with macOS versions earlier than it though.

For reference: Two-factor authentication for Apple ID - Apple Support

I understand you are using this procedure: Macbook Air and App store - Apple Community, but it isn't working.

The inability to access your mail messages using the 10.9.5 version of Mail is likely to be unrelated to it though. Lots of email services require security protocols that Mavericks might not be able to handle, and if that's the case Yosemite will almost certainly exhibit the same problem.

What specific problems are you encountering with the El Capitan upgrade? All I can tell you is that it works. I no longer actively maintain anything older than it though, so my ability to help with Mavericks will be limited at best.