Context. The interstellar medium is the locus of physical processes affecting the evolution of galaxies which drive or are the result of star formation activity, supermassive black hole growth, and feedback. The resulting physical conditions determine the observable chemical abundances that can be explored through molecular emission observations at millimeter and submillimeter wavelengths. Aims. Our goal is to unveiling the molecular richness of the central region of the prototypical nearby starburst galaxy NGC 253 at an unprecedented combination of sensitivity, spatial resolution, and frequency coverage. Methods. We used the Atacama Large Millimeter/submillimeter Array (ALMA), covering a nearly contiguous 289 GHz frequency range between 84.2 and 373.2 GHz, to image the continuum and spectral line emission at 1.6 ''(similar to 28 pc) resolution down to a sensitivity of 30 - 50 mK. This article describes the ALMA Comprehensive High-resolution Extragalactic Molecular Inventory (ALCHEMI) large program. We focus on the analysis of the spectra extracted from the 15 '' (similar to 255 pc) resolution ALMA Compact Array data. Results. We modeled the molecular emission assuming local thermodynamic equilibrium with 78 species being detected. Additionally, multiple hydrogen and helium recombination lines are identified. Spectral lines contribute 5 to 36% of the total emission in frequency bins of 50 GHz. We report the first extragalactic detections of C2H5OH, HOCN, HC3HO, and several rare isotopologues. Isotopic ratios of carbon, oxygen, sulfur, nitrogen, and silicon were measured with multiple species. Concluison. Infrared pumped vibrationaly excited HCN, HNC, and HC3N emission, originating in massive star formation locations, is clearly detected at low resolution, while we do not detect it for HCO+. We suggest high temperature conditions in these regions driving a seemingly "carbon-rich" chemistry which may also explain the observed high abundance of organic species close to those in Galactic hot cores. The L-vib/L-IR ratio was used as a proxy to estimate a 3% contribution from the proto super star cluster to the global infrared emission. Measured isotopic ratios with high dipole moment species agree with those within the central kiloparsec of the Galaxy, while those derived from C-13/O-18 are a factor of five larger, confirming the existence of multiple interstellar medium components within NGC 253 with different degrees of nucleosynthesis enrichment. The ALCHEMI data set provides a unique template for studies of star-forming galaxies in the early Universe.