A million binaries from Gaia eDR3: sample selection and validation of Gaia parallax uncertainties

We construct from Gaia eDR3 an extensive catalogue of spatially resolved binary stars within ≈1 kpc of the Sun, with projected separations ranging from a few au to 1 pc. We estimate the probability that each pair is a chance alignment empirically, using the Gaia catalogue itself to calculate the rate of chance alignments as a function of observables. The catalogue contains 1.3 (1.1) million binaries with >90 per cent (>99 per cent) probability of being bound, including 16 000 white dwarf - main-sequence (WD + MS) binaries and 1400 WD + WD binaries. We make the full catalogue publicly available, as well as the queries and code to produce it. We then use this sample to calibrate the published Gaia DR3 parallax uncertainties, making use of the binary components' near-identical parallaxes. We show that these uncertainties are generally reliable for faint stars (G ≳ 18), but are underestimated significantly for brighter stars. The underestimates are generally $\leq30{{\ \rm per\ cent}}$ for isolated sources with well-behaved astrometry, but are larger (up to ~80 per cent) for apparently well-behaved sources with a companion within ≲4 arcsec, and much larger for sources with poor astrometric fits. We provide an empirical fitting function to inflate published σ_ϖ values for isolated sources. The public catalogue offers wide ranging follow-up opportunities: from calibrating spectroscopic surveys, to precisely constraining ages of field stars, to the masses and the initial-final mass relation of WDs, to dynamically probing the Galactic tidal field.