We present the detection of five planets--Kepler-62b, c, d, e, and f--of size 1.31, 0.54, 1.95, 1.61 and 1.41 Earth radii (R⊕), orbiting a K2V star at periods of 5.7, 12.4, 18.2, 122.4, and 267.3 days, respectively. The outermost planets, Kepler-62e and -62f, are super-Earth-size (1.25 R⊕ < planet radius ≤ 2.0 R⊕) planets in the habitable zone of their host star, respectively receiving 1.2 ± 0.2 times and 0.41 ± 0.05 times the solar flux at Earth's orbit. Theoretical models of Kepler-62e and -62f for a stellar age of ~7 billion years suggest that both planets could be solid, either with a rocky composition or composed of mostly solid water in their bulk.