Being locked out from google account is one of my biggest fears in practical computer usage. Over the years I can't even tell how much I'm sunk into using tiny pieces of google's services, even though for most crucial things I tend to omit google and look for alternatives that are easier to handle in problematic situation. Mail is one of the things I'm most worried about.
Use an IMAP client. It would keep all your mail on your device, so at least you don't have to rely on having uninterrupted access to your account for your old conversations.
Actually, I'm surprised just how many people use email through web interfaces for some reason.