Healthcare is increasingly datafied, and a wide range of actors—patients, clinicians, administrators, policymakers, and industry lobbyists—want to be able to exchange and access health data internationally and use them for an increasing number of purposes. Therefore, competing initiatives aimed at fostering international data integration proliferate, with the proposed European Health Data Space as one of the most prominent examples. But how do legislators conceptualize a health data space? And what could they gain from rethinking the governmental object of this legislation? To explore these questions, we suggest taking the term “data space,” present in the European Health Data Space initiative, and develop it theoretically to establish a vocabulary fit for understanding international data-intensive health environments. Space is a concept with appealing affordances. It is a way of naming a mode of being which is simultaneously symbolic and material, abstract and concrete, social and physical. We show how these affordances of the concept of space can be helpful when exploring new ways of living in cross-border data-intensive healthcare settings. Whereas policy reports often describe data sharing as a matter of providing technical means and legal provisions to “wire together” existing data resources, we argue that data spaces should be understood as sociotechnical constructs enacted through three formative and four experiential dimensions.