This paper develops a well-posedness theory for hyperbolic Maxwell obstacle problems generalizing the result by Duvaut and Lions (1976) [5]. Building on the recently developed result by Yousept (2020) [30], we prove an existence result and study the uniqueness through a local H(curl)-regularity analysis with respect to the constraint set. More precisely, every solution is shown to locally satisfy the Maxwell-Ampère equation (resp. Faraday equation) in the region where no obstacle is applied to the electric field (resp. magnetic field). By this property, along with a structural assumption on the feasible set, we are able to localize the obstacle problem to the underlying constraint regions. In particular, the resulting localized problem does not employ the electric test function (resp. magnetic test function) in the area where the -regularity of the rotation of the electric field (resp. magnetic field) is not a priori guaranteed. This localization strategy is the main ingredient for our uniqueness proof. After establishing the well-posedness, we consider the case where the electric permittivity is negligibly small in the electric constraint region and investigate the corresponding eddy current obstacle problem. Invoking the localization strategy, we derive an existence result under an -boundedness assumption for the electric constraint region along with a compatibility assumption on the initial data. The developed theoretical results find applications in electromagnetic shielding.