Accurate quantification of the ocean carbon sink and its associated uncertainty is critical for guiding international policy efforts and the accurate monitoring, reporting, and verification (MRV) of marine carbon dioxide removal (mCDR) interventions. Here we use error propagation to break down the uncertainty in air-sea CO$_2$ flux into three primary sources: the gas transfer velocity ($k_w$), the solubility (K$_0$), and the difference in partial pressure of CO$_2$ ($\Delta pCO_2$) between the ocean and atmosphere. These are further decomposed into uncertainties from the underlying variables (e.g., temperature and salinity used to calculate K$_0$). We find regional variability in uncertainty, with $\Delta pCO_2$ being the dominant source in subtropical and eastern boundary upwelling systems; gas transfer being the dominant term in the west equatorial Pacific; and solubility dominating uncertainty near the mouths of large rivers. This methodology provides a foundation for a comprehensive quantification of uncertainty and its underlying drivers. The software used in this study is publicly available on GitHub (\url{https://github.com/lgloege/fluxerror}).