Representing glacial-interglacial changes in ocean carbon sequestration remains a major challenge for Earth system models (ESMs). Uncertainties in ocean circulation and biological carbon export are essential causes for model-proxy data mismatch. We quantify the impact of these factors by calibrating the Max Planck Institute-ESM. A shallower and weaker glacial Atlantic meridional overturning circulation (AMOC) than the present day, achieved by decreasing background vertical diffusivity, enables capturing the main features in observed δ13C, 14C and CO32-. Additionally, a new prognostic computation of marine aggregate sinking speeds enhances organic matter export efficiency in high latitudes. Together, the shallower AMOC and a comprehensive sinking scheme substantially improve the model-data comparison and carbon storage in the glacial Atlantic, but not in the Pacific and Indian oceans owing to low organic matter remineralisation. Our results yield that representing the glacial-interglacial ocean carbon storage in ESMs requires both constraining ocean circulation and improved biogeochemical processes.