I recently was given an old ipod touch 2g. Because it can't be updated to iOS 5+, a lot of the new apps don't work for it, including google's new apps. Because I use google's 2-step verification, that meant that putting in my regular google password didn't get me into gmail or calendar. The ipod touch 2g (and most devices, I'm guessing) can be used with google's two-step verification, as it turns out. I've successfully used this approach with Thunderbird as well:
1. Go to your account setting (go to gmail, click your name or picture, click "account")
2. Click "Security"
3. Scroll to the "2-step verification" section
4. Follow "Manage your application specific passwords" link
5. If requested, re-enter your password
6. Follow the directions in the box, "Step 1 of 2: Generate new application-specific password"
6b. (You will put in a name for your device, e.g. ipod touch 2g)
7. Click "Generate password"
8. When you set up your gmail account in the ipod touch 2g (or other device/application), use this application-specific instead of your normal password. (For ipod touch 2g: Setting>Mail, Contacts, Calendars>Add Account...)
9. That's it, good luck!!!