We present the extended GALEX Arecibo SDSS Survey (xGASS), a gas fraction-limited censusof the atomic hydrogen (H I) gas content of 1179 galaxies selected only by stellar mass (M* = 109-1011.5 Mo˙) and redshift (0.01<z<0.05). This includes new Arecibo observations of 208galaxies, for which we release catalogues and HI spectra. In addition to extending the GASSHI scaling relations by one decade in stellar mass, we quantify total (atomic+molecular) coldgas fractions and molecular-to-atomic gas mass ratios, Rmol, for the subset of 477 galaxiesobserved with the IRAM 30 m telescope. We find that atomic gas fractions keep increasingwith decreasing stellar mass, with no sign of a plateau down to logM*/Mo˙ = 9. Total gasreservoirs remain HI-dominated across our full stellar mass range, hence total gas fractionscaling relations closely resemble atomic ones, but with a scatter that strongly correlates withRmol, especially at fixed specific star formation rate. On average, Rmol weakly increases withstellar mass and stellar surface density μ*, but individual values vary by almost two ordersof magnitude at fixed M* or μ*. We show that, for galaxies on the star-forming sequence, variations of Rmol are mostly driven by changes of the HI reservoirs, with a clear dependenceon μ*. Establishing if galaxy mass or structure plays the most important role in regulating thecold gas content of galaxies requires an accurate separation of bulge and disc components forthe study of gas scaling relations.