Can a doctor send you a bill 8 years later because you have no recollection of this service and you have no way of proving that you paid for it?

sure he can, but collecting is another thing. 3 years statute of limitations on collections

Medical bills are considered written contracts. In the majority of states a medical bill is collectible regardless of the designated SOL because a judges will almost always waive SOL laws in such issues. The collection of defaulted medical charges is often possible without the need for litigation as the majority of states also allow for arbitration. The collector is legally required to provide proof of the debt owed upon the request of the alledged debtor.