Extreme disasters have become increasingly common in recent years and pose significant dangers to the integrated energy system’s secure and dependable energy supply. As a vital part of an integrated energy system, the energy storage system can help with emergency rescue and recovery during major disasters. In addition, it can improve energy utilization rates and regulate fluctuations in renewable energy under normal conditions. In this study, the sizing scheme of multi-energy storage equipment in the electric–thermal–hydrogen integrated energy system is optimized; economic optimization in the regular operating scenario and resilience enhancement in extreme disaster scenarios are also considered. A refined model of multi-energy storage is constructed, and a two-layer capacity configuration optimization model is proposed. This model is further enhanced by the integration of a Markov two-state fault transmission model, which simulates equipment defects and improves system resilience. The optimization process is solved using the tabu chaotic quantum particle swarm optimization (TCQPSO) algorithm to provide reliable and accurate optimization results. The results indicate that addressing severe disaster situations in a capacity configuration fully leverages the reserve energy function of energy storage and enhances system resilience while maintaining economic efficiency; furthermore, adjusting the load loss penalty coefficients offers a more targeted approach to the balancing of the system economy and resilience. Thus, new algorithmic choices and planning strategies for future research on enhancing the resilience of integrated energy systems under extreme disaster scenarios are provided.