We report the discovery of two super-Earth-sized planets transiting the bright (V = 8.94, K = 7.07) nearby late G-dwarf HD 3167, using data collected by the K2 mission. The inner planet, HD 3167 b, has a radius of 1.6 R_e and an ultra-short orbital period of only 0.96 days. The outer planet, HD 3167 c, has a radius of 2.9 R_e and orbits its host star every 29.85 days. At a distance of just 45.8 +/- 2.2 pc, HD 3167 is one of the closest and brightest stars hosting multiple transiting planets, making HD 3167 b and c well suited for follow-up observations. The star is chromospherically inactive with low rotational line-broadening, ideal for radial velocity observations to measure the planets masses. The outer planet is large enough that it likely has a thick gaseous envelope which could be studied via transmission spectroscopy. Planets transiting bright, nearby stars like HD 3167 are valuable objects to study leading up to the launch of the James Webb Space Telescope.