Fundamental physical constants are determined from a collection of precision measurements of elementary particles, atoms and molecules. This is usually done under the assumption of the Standard Model~(SM) of particle physics. Allowing for light new physics~(NP) beyond the SM modifies the extraction of fundamental physical constants. Consequently, setting NP bounds using these data, and at the same time assuming the CODATA recommended values for the fundamental physical constants, is not reliable. As we show in this Letter, both SM and NP parameters can be simultaneously determined in a consistent way from a global fit. For light vectors with QED-like couplings, such as the dark photon, we provide a prescription that recovers the degeneracy with the photon in the massless limit, and requires calculations only at leading order in the small new physics couplings. At present, the data show tensions partially related to the proton charge radius determination. We show that these can be alleviated by including contributions from a light scalar with flavor non-universal couplings.