The closet exoplanets to the Sun provide opportunities for detailed characterization of planets outside the Solar System. We report the discovery, using radial velocity measurements, of a compact multiplanet system of super-Earth exoplanets orbiting the nearby red dwarf star GJ 887. The two planets have orbital periods of 9.3 and 21.8 days. Assuming an Earth-like albedo, the equilibrium temperature of the 21.8-day planet is ~350 kelvin. The planets are interior to, but close to the inner edge of, the liquid-water habitable zone. We also detect an unconfirmed signal with a period of ~50 days, which could correspond to a third super-Earth in a more temperate orbit. Our observations show that GJ 887 has photometric variability below 500 parts per million, which is unusually quiet for a red dwarf.