Surgery planning decisions include which operating rooms (ORs) to open, allocation of surgeries to ORs, sequence, and time to start each surgery. They are often made under uncertain surgery durations with limited data that lead to unknown distributional information. Moreover, cost parameters for criteria such as overtime and surgery delays are often difficult or impossible to estimate in practice. In this paper, we formulate distributionally robust (DR) chance constraints on surgery waiting and OR overtime, which recognize practical limitations on data availability and cost parameter accuracy. We use ϕ-divergence measures to build an ambiguity set of possible distributions of random surgery durations, and derive a branch-and-cut algorithm for optimizing a mixed-integer linear programming reformulation based on finite samples of the random surgery durations. We test instances generated from real hospital-based surgery data. The results show computational efficacy of our approaches, and provide insights for DR surgery planning.