We devise and analyze hybrid polyhedral methods of arbitrary order for the approximation of div-curl systems on three-dimensional domains featuring non-trivial topology. The systems we focus on stem from magnetostatics models, and can either be first-order (field formulation) or second-order (vector potential formulation). The well-posedness of our methods essentially relies on topologically general discrete (hybrid) versions of the first and second Weber inequalities. In turn, our error analysis is performed under low regularity assumptions on the solutions. Finally, we provide a comprehensive numerical validation of our methodology.