Society is becoming increasingly dependent on data-rich, “Big Tech” platforms and social networks, such as Facebook and Google. But what happens to our data when these companies close or fail? Despite the high stakes involved, this topic has received only limited attention to date. In this article, we use the hypothetical failure of Facebook as a case study to analyse legal and ethical risks related to the closure of data-rich, Big Tech platforms. Focusing on the EU, we argue that existing governance frameworks are inadequate for addressing these risks and make preliminary recommendations with a view to setting an agenda for future research and policymaking on the demise of Big Tech platforms and data-rich companies more broadly.