Twenty-First Century Courts Act This bill requires that certain information regarding Supreme Court Justices, federal judges, and court proceedings be made publicly available. It also requires the Supreme Court to establish a code of ethics for Justices. If a Justice or judge disqualifies himself or herself from a case, the relevant court must publish on its website a notice explaining the circumstances of the disqualification, with exceptions for health and privacy concerns. In addition, the financial disclosure of each Justice and judge must be available on a public website. Audio recordings of public proceedings before the Supreme Court or a federal court of appeals must be made available online in real time to the extent practicable. The recordings must remain available online for specified lengths of time. Additionally, the Administrative Office of the U.S. Courts must establish a single system for all public federal court records. The system must be publicly accessible at no cost.