The effects of intrinsic fluctuations on bistable dynamics of a chemical system are investigated within the framework of a chemical master equation. Calculations are carried out for a three-variable mesoscopic model of the chemical Chua reaction, a mass-action model that displays the bistable state consisting of a period-1 and chaotic states. Contrary to the deterministic description, the chaotic state can be stable or unstable strongly depending on the system size, indicating a new concept of stability concerning multistable states in mesoscopic dynamics. In addition, the intensity of intrinsic fluctuations for one variable of the system is quite larger than those for the other two variables, implying that the data information obtained by measuring one variable in experiments may not reconstruct the real dynamic behavior of the system. Their implications for ecological and microscopic biological systems are also pointed out.