This paper proposes an optimal planning method for the dual-zero microgrid (DZMG) on an island. The DZMG is the off-grid microgrid that exchanges zero power with entity grids and operates in a net-zero carbon emission mode. A net-zero emission operating strategy is designed considering the positive interaction between CO₂ flow and energy flow. The multi-scale circulation of CO₂ flow is realized by coordinating the carbon capture system, solvent storage tank (SST), and direct air capture (DAC), while the seasonal shift of energy flow is completed by the hydrogen storage system (HSS) and DAC. An optimization planning model for DZMG is developed to size the SST, DAC, and HSS optimally. The net-zero emission target and the net-zero operating strategy are involved in the model to balance the environmental and economic concerns in the planning. Numerical experiments are carried out on two IEEE test systems and a real-world island microgrid to validate the effectiveness and adaptability of the proposed method. Simulation results reveal that the proposed method reduces the planning cost by over 25% compared with the extant zero-carbon-based method. Besides, the economy of the DZMG positively correlates to carbon prices and technological maturity, while inversely relates to fuel prices.