old emails pushing me over the data 15GB limit

I have a Gmail account and I am keeping old emails, in and out, in Outlook all the way back to 2001.  I store them in folders inside my main inbox folder.  I find that all this email is pushing me over google's 15GB free limit.  How can I safely keep all these old emails in Outlook, but detach them somehow from google?  I am not even clear on the terminology here.  I am not concerned about reaching these older emails from "any computer"; only on my primary PC will do.  Also I am not concerned about backing them up as my email database files live within my dropbox folder.

So then, how to disconnect old email folders in Outlook from gmail's servers? 

Am I in the right place to be asking this?

Thanks,

Steve