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Aims. Our aim is to derive carbon isotopic ratios from optically thin tracers in the central regions of the starburst galaxies M 82 and NGC 253. Methods. We present high-sensitivity observations of CCH and two of its 13C isotopologues, C13CH and 13CCH, as well as the optically thin emission from C18O and 13C18O. We assume the column density ratio between isotopologues is representative of the 12C/13C isotopic ratio. Results. From CCH, lower limits to the 12C/13C isotopic ratio of 138 in M 82, and 81 in NGC 253, are derived. Lower limits to the 12C/13C ratios from CO isotopologues support these.13C18O is tentatively detected in NGC 253, which is the first reported detection in the extragalactic ISM. Based on these limits, we infer ratios of 16O/18O > 350 and > 300 in M 82 and NGC 253, respectively, and 32S/34S > 16 in NGC 253. The derived CCH fractional abundances toward these galaxies of ≲ 1.1 × 10-8 agree well with those of molecular clouds in the Galactic disk. Conclusions. Our lower limits to the 12C/13C ratio from CCH are a factor of 2−3 larger than previous limits. The results are discussed in the context of molecular and nucleo-chemical evolution. The high 12C/13C isotopic ratio of the molecular ISM in these starburst galaxies suggest that the gas has been recently accreted toward their nuclear regions