We introduce a quantum optimal transport analogue of Wasserstein barycenters for quantum states on separable Hilbert spaces. Our framework allows possibly unbounded transport cost operators and optional energy constraints on the barycenter. We establish existence of barycenters for two broad classes of costs, namely compact-resolvent costs and quadrature-generated costs, and prove a strong duality formula in terms of operator-valued Kantorovich potentials. The latter class includes the canonical quadratic costs arising in continuous-variable quantum optimal transport and provides a unified treatment of the density-coupling and quantum-channel formulations of the 2-quantum Wasserstein problem. We then study Gaussian input states. We show that Gaussianization cannot increase the barycenter objective, so Gaussian inputs always admit a Gaussian barycenter and the infinite-dimensional problem reduces to a finite-dimensional convex optimization over covariance matrices. For faithful Gaussian inputs, we further prove a rigidity result for the density-coupling formulation: the barycenter is unique among all quantum states and the unique minimizer is Gaussian.