Do you live in the USA? If so, your ZIP Code is 5 digits long and 00000 is not a legitimate ZIP Code in any state or territory of the US. If you live outside the USA, your country's postal administration controls Postal Codes. Likewise to the USA, all zeroes is not a legitimate postal code anywhere in the world.
The bottom line is that you must use a real, verifiable postal address when registering your Apple ID regardless of whether that address is the same as the billing address of your payment method (or empty payment method) or not.