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 similar to 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 similar to 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. ; German Research Foundation (DFG) FOR2544 JE 701/3-1 German Research Foundation (DFG) SPP 1992 JE 701/5-1 Science & Technology Facilities Council (STFC) ST/P000584/1 ST/T000295/1 ST/P000592/1 Spanish Agencia Estatal de Investigacion AYA2017-89637R AYA2016-79425-C3-3-S ESP2017-87676-C5-2-R ESP2017-87143R PGC2018-098153-B-C31 ESP2016-80435-C2-2-R Centre of Excellence "Severo Ochoa" Instituto de Astrofisica de Andalucia SEV-2017-0709 Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) CONICYT FONDECYT 3180405 Swiss National Science Foundation (SNSF) CONICYT/PFCHA-Doctorado Nacional, Chile 21140646 Spanish Government ESP2016-80435-C2-1-R PGC2018-098153-B-C33 European Union (EU) ESP2016-80435-C2-1-R PGC2018-098153-B-C33 Generalitat de Catalunya/CERCA program Science & Technology Facilities Council (STFC) ST/M001008/1 Australian Research Council DP0774000 DP130102695 DP170103491 NASA through Hubble Fellowship - Space Telescope Science Institute HST-HF2-51399.001 Ministerio de Ciencia, Innovacion y Universidades Ramon y Cajal fellowship RYC-2017-22489 Science & Technology Facilities Council (STFC) ST/P000592/1 National Aeronautics & Space Administration (NASA) NAS5-26555