Legally? Yes. Politically? Almost certainly not. Australia's Constitution sets remarkably few formal qualifications for prime minister—just membership in Parliament (House or Senate) and the ability to command the confidence of the House of Representatives. Nowhere does it mention criminal history.
Section 64 requires ministers to be (or become) parliamentarians. Section 44 lists disqualifications for parliament—citizenship, bankruptcy, office of profit under the Crown, treason—but not criminal convictions per se. A conviction only triggers disqualification if it carries a potential sentence of one year or more and the person is actually serving that sentence (or awaiting sentencing). A past conviction, served sentence, or non-custodial penalty? No constitutional bar.
Australia inherits the Westminster convention: the PM must command the lower house's confidence. In practice, this means:
A "serious criminal record" (violence, corruption, fraud, sexual offenses) would:
Minor, historic, or spent convictions might be survivable with full disclosure and rehabilitation evidence. But the premise—a PM with a significant record—remains theoretical. The Constitution doesn't forbid it; the electorate effectively does.