Building and operating cutting-edge, custom-designed research facilities presents unique challenges, where prior designs and experience may not be a reliable guide and unprecedented modes of failure can never be fully anticipated. In 2020, the National Science Foundation's telescope at the Arecibo Observatory in Puerto Rico collapsed, impacting the work of the National Astronomy and Ionosphere Center. Failure Analysis of the Arecibo Observatory 305-Meter Telescope Collapse analyzes the causes of the collapse through extensive review of prior forensic investigations, information gathering from employees at Arecibo Observatory, study of relevant research, consultations with other experts, and examination of structural analyses, engineering plans, inspection reports, photographs, and repair proposals. This report presents lessons learned and makes recommendations to help ensure the safe operation of other unique, critical science facilities.