The evolved massive binary star, Eta Car, underwent eruptive mass loss events that produced the complex bipolar Homunculus nebula and a torus-like equatorial structure that harbors 10s of solar masses of nitrogen-rich, oxygen- and carbon-depleted gas and dust. A significant molecular component to the gas should be present, but detections have been observationally challenged by limited access to the needed wavelengths and the intense thermal continuum. A far-infrared spectral survey of the atomic and rotational molecular transitions was carried out with the spectrometers onboard the Herschel telescope, revealing a rich spectrum of broad emission lines originating in the ejecta surrounding the central binary. We present the scans obtained with the Herschel PACS (40 to 190 microns) and SPIRE (193 to 689 microns) instruments. Velocity profiles of selected PACS lines correlate well with known substructures: HI in the central core; NH and weak [CII] emission components that spatially correlate with known structures within the Homunculus and its extended skirt; and spatially-extended velocity-shifted [N II] emissions in fast moving structures outside the Homunculus. We have identified transitions from [OI], the HI Rydberg states, and 19 separate light C- and O-bearing molecules including CO, CH, CH+, and OH, and a wide set of N-bearing NH, NH+, N2H+, NH2, NH3, HCN, HNC, CN, and N2H+. About half of these are new detections in Eta Car, and most are unprecedented for any evolved massive star environment. A very low ratio [12C/13C] < 4 is estimated from five molecules and their isotopologues. We demonstrate that non-LTE effects due to the strong continuum are significant in Eta Car. Abundance patterns are consistent with line formation in regions of carbon and oxygen depletions and nitrogen enhancements, reflecting an evolved state of the erupting star or merger with efficient rotational transport of CNO-processed material to the outer layers. The results offer many opportunities for further observational and theoretical investigations of the molecular chemistry under extreme conditions around massive stars in their final stages of evolution.