The building system is one of key energy consumption sector in the market, and low-carbon building will make a significant contribution for the worldwide carbon emission reduction. The multiple energy systems including renewable generations, hydrogen energy and energy storage is the perspective answer to the net-zero building system. However, the research gap lies in the synergy power management among the renewable, flexible loads, batteries and hydrogen energy systems, and at the same time, taking the unique characteristic of different energy sectors into account by power managing. This paper proposed the power management approach based on the game theory, by which, the different characteristics of the energy players are described via creating the competing relationship against net-zero emission objective so that to achieve the power synergy. Under the proposed power management method, the hydrogen and battery hybrid system including the fuel cell, electrolyzer and battery is designed and investigated as to unlock the power management regions and control constraints within the building system. Particularly, for the hydrogen system within the hybrid system, the safe and long-lifetime operation is considered respectively by high-efficiency and pressure constraints within the power management. Simulation results show that, providing the same energy storage services for the building system, the fuel cell with the proposed power management method sustains for 9.9 years, much longer than that of equivalent consumption minimization (4.98), model predictive control (4.61) and rule-based method (7.69). Moreover, the maximum tank temperature of the hydrogen tank is reduced by 3.4 K and 2.9 K compared with consumption minimization strategy and model predictive control. Also, the real-time of the proposed power management is verified by a scaled-down experiment platform.