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Buildings produce a significant share of greenhouse gas (GHG) emissions, making homes and businesses a major factor in climate change. To address this critical challenge, this paper explores achieving net-zero emission through the carbon-aware optimal scheduling of the multi-energy building integrated energy systems (BIES). We integrate advanced technologies and strategies, such as the carbon capture system (CCS), power-to-gas (P2G), carbon tracking, and emission allowance trading, into the traditional BIES scheduling problem. The proposed model enables accurate accounting of carbon emissions associated with building energy systems and facilitates the implementation of low-carbon operations. Furthermore, to address the challenge of accurately assessing uncertainty sets related to forecasting errors of loads, generation, and carbon intensity, we develop a learning-based robust optimization approach for BIES that is robust in the presence of uncertainty and guarantees statistical feasibility. The proposed approach comprises a shape learning stage and a shape calibration stage to generate an optimal uncertainty set that ensures favorable results from a statistical perspective. Numerical studies conducted based on both synthetic and real-world datasets have demonstrated that the approach yields up to 8.2% cost reduction, compared with conventional methods, in assisting buildings to robustly reach net-zero emissions.