With the high penetration of renewable energy integrated into the grid, short-term power imbalances in power systems intensify, posing challenges to frequency security. Deploying energy storage is regarded as a critical means to address this challenge. In existing energy storage planning methods that consider frequency security, power fluctuations are usually presupposed when deriving frequency security constraints, leading to overly conservative results; moreover, the nonlinear frequency constraints become computationally complex under uncertainty. To this end, a multi-objective distributionally robust energy storage capacity planning method is proposed. Typical daily scenarios are generated through clustering, the nonlinear frequency security constraints are transformed into a mixed-integer programming formulation via scenario enumeration, and distributionally robust optimization is employed to handle power uncertainty. Pareto solution sets with frequency security satisfied at different confidence levels are provided through case studies, and the impacts of key factors such as frequency security constraints and green objective weights on energy storage configuration are further analyzed. It is demonstrated that the proposed method can effectively coordinate economic, green, and security objectives, offering decision support for energy storage planning in regional power systems.