With their European strategy for data, the EU aims to improve access to and promote the exchange of data in order to create a European single market. Key instruments in this effort are Common European Data Spaces intended to remove legal and practical barriers to data sharing. This article analyzes selected legal obstacles to data sharing, in particular a lack of oversight, uncertainties regarding data protection law, and regulatory fragmentation. The focus is on the European Health Data Space (EHDS), the first implementation of such a data space. Using the EHDS as a case study, the article examines the extent to which Common European Data Spaces are suitable for enabling data sharing, what limitations exist, and under what conditions a transfer to other sectors is conceivable.