Variable levels of methane in the martian atmosphere have eluded explanation partly because the measurements are not repeatable in time or location. We report in situ measurements at Gale crater made over a 5-year period by the Tunable Laser Spectrometer on the Curiosity rover. The background levels of methane have a mean value 0.41 +/- 0.16 parts per billion by volume (ppbv) (95% confidence interval) and exhibit a strong, repeatable seasonal variation (0.24 to 0.65 ppbv). This variation is greater than that predicted from either ultraviolet degradation of impact-delivered organics on the surface or from the annual surface pressure cycle. The large seasonal variation in the background and occurrences of higher temporary spikes (similar to 7 ppbv) are consistent with small localized sources of methane released from martian surface or subsurface reservoirs.