Believe it or not this is actually documented. Maybe that's not the most appropriate place to mention something like this, but it is there:

In the Google payments center, you can permanently close your Google payments profile. Closing your Google payments profile stops payment for all Google services associated with the profile, and removes your transaction and payment information permanently. You can't reopen a closed payments profile.

Here are the steps (props to this guy), but before proceeding keep in mind that you won't be able to add another payment method afterwards in the same billing account because of a bug.

1. Go to https://payments.google.com/ (Yes, you have to go to a completely different website).
2. Go to Settings.
   
3. Scroll all the way down to "Payments profile status" and click on "Close payments profile".
   
4. Select an option in the popup that opens and click "Continue".
   
5. "Close payments profile".
   

If you go back to Google Cloud and reload, this is how the screen with your payment method will look like:

As you can see, not only did your payment method dissappear, but also the rest of the UI and even the button to add another payment method. This implies that the developers didn't account for this specific use case and now it doesn't work properly.

In fact, if you go now to Account management you will notice that the billing account closed automatically and you can't reopen it: "You can’t reopen this billing account because this account is not in good standing.".