Americans needing to have this document should consult an attorney or the services of a notary.
"In the past, the U.S. Embassy allowed American citizens to swear such an affidavit of eligibility to marry before a U.S. consular officer. The Embassy discontinued this practice several years ago, however, because local officials were interpreting these documents as meaning that the Embassy had actually verified the content of the citizens’ statements, when in fact the consular officer was merely attesting to the fact that the individual in question had made the statement." Source: U.S. Embassy
In Australia, this is called a Certificate of No Record and can be obtained from the Registry of Births, Deaths and Marriages in Australia.