All your address are belong to Google
So where is this little feat accomplished? If you sign in to "My account" on Google, and hit "edit" by personal information, a dialog that allows you to add additional email addresses should be visible below your usual info. If you try to add other Google addresses you will quickly be denied, convenient as that would otherwise be. Once you add an acceptable address, you'll have to verify your account via a little confirmation email sent to your non-Gmail account. Then you'll be set.
Functionality at this point, in terms of synchronicity, still seems limited to Google Calendar. But hopefully this will be expanded to other Google services that allow sharing and invites. Considering that we can already use Gmail to check and send email from non-Gmail accounts, this brings us just a little closer to having our non-Google accounts behave as if they were.