This issue is most likely caused by attempting to enroll using the web site, rather than the Apple Developer app.
To prevent this issue, use the Apple Developer app on an iOS device to enroll in the developer program, not the website.
If you are already encountering this, I cannot stress enough how important it is to contact Apple Developer Support as suggested in the recommended answer - Apple's main tech support and billing support have no visibility into this system and cannot transfer you to the developer support.
In my case the enrollment needed to be deleted. The support rep then had to refresh his screen several times to see and clear an error so I could re-enroll (using the same info) through the app. The rep was then able to quickly validate my enrollment (took maybe 2 minutes) and finally, the payment worked!!
The key to resolving this is getting to the correct department, which you cannot do through the main support line.