Unknown-input observers (UIOs) allow for estimation of the states of an LTI system without knowledge of all inputs. In this letter, we provide a novel data-driven UIO based on behavioral system theory and the result known as Fundamental Lemma proposed by Jan Willems and coworkers. We give necessary and sufficient conditions on the data collected from the system for the existence of a UIO providing asymptotically converging state estimates, and propose a purely data-driven algorithm for their computation. Even though we focus on UIOs, our results also apply to the standard case of completely known inputs. As an example, we apply the proposed method to distributed state estimation in DC microgrids.