  We construct from Gaia eDR3 an extensive catalog of spatially resolved binary
stars within $\approx$ 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 catalog itself to calculate the rate of
chance alignments as a function of observables. The catalog contains 1.3 (1.1)
million binaries with >90% (>99%) probability of being bound, including 16,000
white dwarf -- main sequence (WD+MS) binaries and 1,400 WD+WD binaries. We make
the full catalog 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
($G\gtrsim 18$), but are underestimated significantly for brighter stars. The
underestimates are generally $\le 30\%$ for isolated sources with well-behaved
astrometry, but are larger (up to 80%) for apparently well-behaved sources with
a companion within $\lesssim 4$ arcsec, and much larger for sources with poor
astrometric fits. We provide an empirical fitting function to inflate published
$\sigma_{\varpi}$ values for isolated sources. The public catalog 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 white dwarfs, to dynamically probing the Galactic tidal field.
