The distribution of planet sizes encodes details of planet formation and evolution. We present the most precise planet size distribution to date based on Gaia parallaxes, Kepler photometry, and spectroscopic temperatures from the California-Kepler Survey. Previously, we measured stellar radii to 11% precision using high-resolution spectroscopy; by adding Gaia astrometry, the errors are now 3%. Planet radius measurements are, in turn, improved to 5% precision. With a catalog of ~1000 planets with precise properties, we probed in fine detail the gap in the planet size distribution that separates two classes of small planets, rocky super-Earths and gas-dominated sub-Neptunes. Our previous study and others suggested that the gap may be observationally under-resolved and inherently flat-bottomed, with a band of forbidden planet sizes. Analysis based on our new catalog refutes this; the gap is partially filled in. Two other important factors that sculpt the distribution are a planet's orbital distance and its host-star mass, both of which are related to a planet's X-ray/UV irradiation history. For lower-mass stars, the bimodal planet distribution shifts to smaller sizes, consistent with smaller stars producing smaller planet cores. Details of the size distribution including the extent of the "sub-Neptune desert" and the width and slope of the gap support the view that photoevaporation of low-density atmospheres is the dominant evolutionary determinant of the planet size distribution.