The cosmic-ray ionization rate (CRIR) is a key parameter in understanding the physical and chemical processes in the interstellar medium. Cosmic rays are a significant source of energy in star formation regions, impacting the physical and chemical processes that drive the formation of stars. Previous studies of the circum-molecular zone of the starburst galaxy NGC 253 have found evidence for a high CRIR value: 103-106 times the average CRIR within the Milky Way. This is a broad constraint, and one goal of this study is to determine this value with much higher precision. We exploit ALMA observations toward the central molecular zone of NGC 253 to measure the CRIR. We first demonstrate that the abundance ratio of H3O+ and SO is strongly sensitive to the CRIR. We then combine chemical and radiative transfer models with nested sampling to infer the gas properties and CRIR of several star-forming regions in NGC 253 from emission from their transitions. We find that each of the four regions modeled has a CRIR in the range (1-80) × 10-14 s-1 and that this result adequately fits the abundances of other species that are believed to be sensitive to cosmic rays, including C2H, HCO+, HOC+, and CO. From shock and photon-dominated/X-ray dominated region models, we further find that neither UV-/X-ray-driven nor shock-dominated chemistry is a viable single alternative as none of these processes can adequately fit the abundances of all of these species.