You don't need to back up the Google Authenticator app/data as you can create a list of 'Backup codes' which you can use to log in without requiring a authentication code on the same page that you configure 2-step authentication.
Why print or download backup codes?
Backup codes are especially useful for people who travel, have problems receiving SMS or voice calls, or cannot use the Google Authenticator mobile app.
Recommendation: You should print or download backup codes
Store these in a safe place (or print them out) and if you lose your phone you can use one of these codes to log into your account and set up a new device with the Authenticator app.
Whilst this applies to Google's 2-step-authentication, any other sites you have configured to use the Google Authenticator app should offer a similar option, or another way to receive codes (e.g. Facebook supports Google Authenticator, their own app and sms as methods to receive codes).