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The impact of human activity on the climate is a major global challenge that affects human well-being. Buildings are a major source of energy consumption and carbon emissions worldwide, especially in advanced economies such as the United States. As a result, making grids and buildings sustainable by reducing their carbon emissions is emerging as an important step toward societal decarbonization and improving overall human well-being. While prior work on demand response methods in power grids and buildings has targeted peak shaving and price arbitrage in response to price signals, it has not explicitly targeted carbon emission reductions. In this paper, we analyze the flexibility of building loads to quantify the upper limit on their potential to reduce carbon emissions, assuming perfect knowledge of future demand and carbon intensity. Our analysis leverages real-world demand patterns from 1000+ buildings and carbon-intensity traces from multiple regions. It shows that by manipulating the demand patterns of electric vehicles, heating, ventilation, and cooling (HVAC) systems, and battery storage, we can reduce carbon emissions by 26.93% on average and by 54.90% at maximum. Our work advances the understanding of sustainable infrastructure by highlighting the potential for infrastructure design and interventions to significantly reduce carbon footprints, benefiting human well-being.