We characterize the metallicities and physical properties of cool, photoionized gas in a sample of 152 z<=1 strong Ly{alpha} forest systems (SLFSs, absorbers with 15<logN_HI_<16.2). The sample is drawn from our Cosmic Origins Spectrograph (COS) circumgalactic medium compendium (CCC), an ultraviolet survey of HI-selected circumgalactic gas around z<=1 galaxies that targets 261 absorbers with 15<logN_HI_<19. We show that the metallicity probability distribution function of the SLFSs at z<=1 is unimodal, skewed to low metallicities with a mean and median of [X/H]=-1.47 and -1.18dex. Very metal-poor gas with [X/H]{<}-1.4 represents about half of the population of absorbers with 15<logN_HI_<=18, while it is rare at higher N_HI_. Thus, there are important reservoirs of primitive (though not pristine) diffuse ionized gas around z<=1 galaxies. The photoionized gas around z<=1 galaxies is highly inhomogeneous based on the wide range of metallicities observed (-3<=[X/H]<=+0.4) and the fact that there are large metallicity variations (factors of 2 to 25) for most of the closely spaced absorbers ({Delta}v<=300km/s) along the same sightlines. These absorbers show a complex evolution with redshift and HI column density, and we identify subtle cosmic evolution effects that affect the interpretation of metallicity distributions and comparison with other absorber samples. We discuss the physical conditions and cosmic baryon and metal budgets of the CCC absorbers. Finally, we compare the CCC results to recent cosmological zoom simulations and explore the origins of the 15<logN_HI_<19 absorbers within the Evolution and Assembly of GaLaxies and their Environments (EAGLE) high-resolution simulations.