We develop a two-stage stochastic program for energy and reserve dispatch of a joint power and gas system with a high penetration of renewables. Data-driven distributionally robust chance constraints ensure that there is no load shedding and renewable spillage with high probability. We solve this problem efficiently using conditional value-at-risk approximations and linear decision rules. Out-of-sample experiments show that this model dominates the corresponding stochastic program without chance constraints that models the effects of load shedding and renewable spillage explicitly.