We report the likely identification of a substantial population of massive M 1011 M galaxies at z 4 with suppressed star formation rates (SFRs), selected on rest-frame optical to near-IR colors from the FourStar Galaxy Evolution Survey (ZFOURGE). The observed spectral energy distributions show pronounced breaks, sampled by a set of near-IR medium-bandwidth filters, resulting in tightly constrained photometric redshifts. Fitting stellar population models suggests large Balmer/4000 breaks, relatively old stellar populations, large stellar masses, and low SFRs, with a median specific SFR of 2.9 ± 1.8 × 10-11 yr-1. Ultradeep Herschel/PACS 100 μm, 160 μm and Spitzer/MIPS 24 μm data reveal no dust-obscured SFR activity for 15/19(79%) galaxies. Two far-IR detected galaxies are obscured QSOs. Stacking the far-IR undetected galaxies yields no detection, consistent with the spectral energy distribution fit, indicating independently that the average specific SFR is at least 10 × smaller than that of typical star-forming galaxies at z 4. Assuming all far-IR undetected galaxies are indeed quiescent, the volume density is 1.8 ± 0.7 × 10 -5 Mpc-3 to a limit of log 10 M/M ≥ 10.6, which is 10 × and 80 × lower than at z = 2 and z = 0.1. They comprise a remarkably high fraction (35%) of z 4 massive galaxies, suggesting that suppression of star formation was efficient even at very high redshift. Given the average stellar age of 0.8 Gyr and stellar mass of 0.8 × 10 11 M, the galaxies likely started forming stars before z = 5, with SFRs well in excess of 100 M yr-1, far exceeding that of similarly abundant UV-bright galaxies at z ≥ 4. This suggests that most of the star formation in the progenitors of quiescent z 4 galaxies was obscured by dust.