We present submillimeter spectra of the (proto-)super star cluster (SSC) candidates in the starbursting center of the nearby galaxy NGC 253 identified by Leroy et al. The 2.5 pc resolution of our Atacama Large Millimeter/submillimeter Array cycle 3 observations approaches the size of the SSCs and allows for the study of physical and chemical properties of the molecular gas in these sources. In the 14 SSC sources and in the frequency ranges 342.0-345.8 GHz and 353.9-357.7 GHz, we detect 55 lines belonging to 19 different chemical species. The SSCs differ significantly in chemical complexity, with the richest clusters showing 19 species and the least complex showing four species. We detect HCN isotopologues and isomers (H13CN, HC15N, H15NC), abundant HC3N, SO and S18O, SO2, and H2CS. The gas ratios CO/HCN, CO/HCO+ are low, ∼1-10, implying high dense gas fractions in the SSCs. Line ratio analyses suggests chemistry consistent with photon-dominated regions and mechanical heating. None of the SSCs near the galaxy center show line ratios that imply an X-ray-dominated region, suggesting that heating by any (still unknown) active galactic nucleus does not play a major role. The gas temperatures are high in most sources, with an average rotational temperature of ∼130 K in SO2. The widespread existence of vibrationally excited HCN and HC3N transitions implies strong infrared radiation fields, potentially trapped by a greenhouse effect due to high continuum opacities.