We present Keck-MOSFIRE H and K spectra for a sample of 24 candidate quiescent galaxies (QGs) at 3<z<4, identified from UVJ colors and photometric redshifts in the ZFOURGE and 3DHST surveys. We obtain spectroscopic redshifts for half of the sample, using absorption or emission lines, and confirm the high accuracy of the photometric redshifts with a median error of 1.2%. Two galaxies turn out to be dusty objects at lower redshifts (z<2.5), and are the only two detected in the sub-mm with ALMA. High equivalent-width [OIII] was observed in two galaxies, contributing up to 30% of the K-band flux and mimicking the colors of an old stellar population. This implies a failure rate of only 20% for the UVJ selection at these redshifts. Balmer absorption was identified in 4 of the brighest galaxies, confirming the absence of OB stars. Modeling all QGs with a wide range of star-formation histories, we find sSFR a factor of 10 below the main sequence (MS) for all but one galaxy, and less than 0.01 Gyr$^{-1}$ for half of the sample. This is consistent with the H$beta$ and [OII] luminosities, and the ALMA non-detections. We then find that these QGs have quenched on average 300 Myr before observation, between z=3.5 and 5, and that they formed at z~5.5 with a mean SFR~300 Msun/yr. Considering an alternative selection of QGs based solely on the sSFR from SED modeling, we find that galaxies a factor 10 below the MS are 40% more numerous than UVJ-quiescent galaxies, implying that the UVJ selection is pure but incomplete. Current models fail at reproducing our observations and underestimate either the number density of QGs by more than an order of magnitude or the duration of their quiescence by a factor two. Overall, these results confirm the existence of an unexpected population of QGs at z>3, and offer the first insights on their formation history. [abridged]
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