We study control of constrained linear systems with only partial statistical information about the uncertainty affecting the system dynamics and the sensor measurements. Specifically, given a finite collection of disturbance realizations drawn from a generic distribution, we consider the problem of designing a stabilizing control policy with provable safety and performance guarantees despite the mismatch between the empirical and true distributions. We capture this discrepancy using Wasserstein ambiguity sets, and we formulate a distributionally robust (DR) optimal control problem, which provides guarantees on the expected cost, safety, and stability of the system. To solve this problem, we first present new results for Wasserstein DR optimization of quadratic objectives with additional bounded support constraints, showing that strong duality holds under mild conditions. Then, by combining our results with the system-level parametrization of linear feedback policies, we show that the design problem can be reduced to a semidefinite program. We present numerical simulations to validate the effectiveness of our approach and to highlight the value of centering the ambiguity set at the empirical distribution for control design.