Open Science is a founding principle of ELIXIR, a pan-European research infrastructure for life science data, with 21 Member countries plus the European Molecular Biology Laboratory. The mission of ELIXIR is to coordinate bioinformatics resources so that they form a single, integrated and pan-European infrastructure, which can be used freely by academic and private-sector researchers across the globe. As a recipient of public and charitable funding, ELIXIR must demonstrate its value, and the need to produce evidence in support of this is intensifying. Our practice-led journey towards demonstrating public value is articulated around five main challenges and, for each, we present our pragmatic approach for tackling it. We begin by showing how we are working towards demystifying what research infrastructures do. We then shed light on the sort of evidence our funders and other stakeholders are asking us for, how this evidence varies in nature and scope, and our tactics to satisfy them. We follow-on by providing our thoughts on possible barriers and solutions to embedding impact evaluation in our activities. Finally, we provide lessons learned, which we believe are sufficiently transferable and will be inspirational to other research infrastructures as they embark on their own journeys to demonstrate public value.