Emission from the CO molecule at λ = 2.6 mm has been observed at 1412 positions in 300 galaxies using the 14 m telescope of the Five College Radio Astronomy Observatory (HPBW = 45"); these data comprise the FCRAO Extragalactic CO Survey. In this paper we describe the galaxy sample, present the data, and determine global CO fluxes and radial distributions for the galaxies in the Survey. Future papers will deal with the data analysis, both with regard to the global properties of galaxies and the radial distributions within them. CO emission was detected in 236 of the 300 Survey galaxies for an overall detection rate of 79%; among the 52 Sc galaxies in the Survey, the detection rate was as high as 96%. most of the 193 galaxies observed in multiple positions exhibit CO distributions which peak at the center. However, a small number (10-primarily Sb galaxies) exhibit CO rings at 45" resolution, and a similar number (18-primarily Sc galaxies) have CO distributions which peak on one side of the center. We derive CO isophotal diameters for 151 galaxies and find the mean ratio of CO to optical isophotal diameters to be 0.5. We also find a trend along the Hubble sequence such that the mean ratio of CO to optical isophotal diameters is smallest among the early-type spirals (SO/a, Sa, and Sab) and the mean ratio increases for Sb, Sbc, and Sc galaxies, finally decreasing among the later types. Comparison of the global fluxes we derive for the Survey galaxies with independent measurements from the literature indicates that the global fluxes we derive are accurate to ~40%.