The infrared-luminous galaxy NGC 3256 is a classic example of a merger-induced nuclear starburst system. We find here that it is the most X-ray-luminous star-forming galaxy yet detected (L0.5-10keV = 1.6 x 10(42) ergs s(-1)). Long-slit optical spectroscopy and a deep, high-resolution ROSAT X-ray image show that the starburst is driving a "superwind" which accounts for similar to 20% of the observed soft X-ray emission. Analysis of X-ray spectral data from ASCA indicates this gas has a characteristic temperature of kT approximate to 0.3 keV. Our model for the broadband X-ray emission of NGC 3256 contains two additional components: a warm thermal plasma (kT approximate to 0.8 keV) associated with the central starburst, and a hard power-law component with an energy index of alpha(x) approximate to 0.7. We discuss the energy budget for the two thermal plasmas and find that the input of mechanical energy from the starburst is more than sufficient to sustain the observed level of emission. We also examine possible origins for the power-law component, concluding that neither a buried AGN nor the expected population of high-mass X-ray binaries can account for this emission. Inverse Compton scattering, involving the galaxy's copious flux of infrared photons and the relativistic electrons produced by supernovae, is likely to make a substantial contribution to the hard X-ray flux. Such a model is consistent with the observed radio and IR fluxes and the radio and X-ray spectral indices. We explore the role of X-ray-luminous starbursts in the production of the cosmic X-ray background radiation. The number counts and spectral index distribution of the faint radio source population, thought to be dominated by star-forming galaxies, suggest that a significant fraction of the hard X-ray background could arise from starbursts at moderate redshift.