Far-ultraviolet (FUV; ~1200-2000{AA}) spectra are fundamental to our understanding of star-forming galaxies, providing a unique window on massive stellar populations, chemical evolution, feedback processes, and reionization. The launch of the James Webb Space Telescope will soon usher in a new era, pushing the UV spectroscopic frontier to higher redshifts than ever before; however, its success hinges on a comprehensive understanding of the massive star populations and gas conditions that power the observed UV spectral features. This requires a level of detail that is only possible with a combination of ample wavelength coverage, signal-to-noise, spectral-resolution, and sample diversity that has not yet been achieved by any FUV spectral database. We present the Cosmic Origins Spectrograph Legacy Spectroscopic Survey (CLASSY) treasury and its first high-level science product, the CLASSY atlas. CLASSY builds on the Hubble Space Telescope (HST) archive to construct the first high-quality (S/N_1500{AA}_>~5/resel), high-resolution (R~15000) FUV spectral database of 45 nearby (0.002<z<0.182) star-forming galaxies. The CLASSY atlas, available to the public via the CLASSY website, is the result of optimally extracting and coadding 170 archival+new spectra from 312 orbits of HST observations. The CLASSY sample covers a broad range of properties including stellar mass (6.2<logM_*_(M_{sun}_)<10.1), star formation rate (-2.0<logSFR(M_{sun}_/yr)<+1.6), direct gas-phase metallicity (7.0<12+log(O/H)<8.8), ionization (0.5<O_32_<38.0), reddening (0.02<E(B-V)<0.67), and nebular density (10<n_e_(cm-3)<1120). CLASSY is biased to UV-bright star-forming galaxies, resulting in a sample that is consistent with the z~0 mass-metallicity relationship, but is offset to higher star formation rates by roughly 2dex, similar to z>~2 galaxies. This unique set of properties makes the CLASSY atlas the benchmark training set for star-forming galaxies across cosmic time.