A multisource energy storage system (MESS) among electricity, hydrogen and heat networks from the energy storage operator's prospect is proposed in this article. First, the framework and device model of MESS is established. On this basis, a multiobjective optimal dispatch strategy of MESS is proposed. Considering the influence of time-of-use price, our dispatch strategy can achieve the unified effect of economy, efficiency, and stationarity. Ultimately, the proposed dispatch strategy is applied to a MESS to study the characteristic of the economic benefits, the influence of energy loss, and power fluctuation, etc. The results show that the MESS can be profitable through our optimized dispatch strategy. And the sale of hydrogen and heat can become the main income of the MESS. Furthermore, we find energy loss is the main challenge for MESS development and application. Considering the economy and efficiency of the system, the proposed dispatch strategy can achieve the optimal comprehensive benefits from the energy storage operator's prospect and support the sustainable development of the MESS.