We report the results of structural and physical studies on the MoReRuCx medium-entropy alloys (MEAs) with x varying from 0 to 0.20. In the whole x range, a single hexagonal phase is formed. The variation of lattice parameters indicates that carbon atoms are located at the interstitial site for x ≤ 0.10 while occupy both the interstitial and substitutional sites at higher x values, which is corroborated by the x-ray photonelectron spectroscopy (XPS) results. Moreover, the resistivity, magnetic and thermodynamic measurements evidence that these MEAs are bulk fully gapped superconductors. In particular, the transition temperature Tc displays a dome-like dependence on carbon content with a maximum of 9.62 K at x = 0.05, whereas a maximal zero-temperature upper critical field Bc2(0) of 10.5 T is achieved at x = 0.10. For all MEAs, the specific heat data reveal the existence of an inhomogeneous superconducting state, which is discussed in relation to the chemical inhomogeneity and strong atomic disorder. In addition, we show that the key factors that control superconductivity are essentially the same for both MoReRuCx and ReCx, despite their very different Tc values. Our work not only provides the first series of interstitial MEAs with a hexagonal structure, but also demonstrates a feasible way to tune the superconductivity in multicomponent alloys.