We present the DeepDive program, in which we obtained deep (1 − 3 hours) JWST/NIRSpec G235M/F170LP spectra for ten primary massive (log(M ★/M ⊙) = 10.8 − 11.5) quiescent galaxies at z ∼ 3 − 4. A novel reduction procedure was used to extend the nominal wavelength coverage of G235M beyond Hα and [N II] at z ∼ 4, revealing weak, narrow Hα lines indicative of low star formation rates (SFR ∼ 0 − 5 M ⊙ yr−1). Two out of ten primary targets have broad Hα lines, indicating the presence of active galactic nuclei. We also conducted an archival search of quiescent galaxies observed with NIRSpec gratings in the DAWN JWST Archive, which provides a statistical context for interpreting the DeepDive targets. This archival search provided a spectroscopic sample of 126 quiescent galaxies spanning 1 < z < 5, selected by high D n 4000, UVJ color, or low specific star formation rate (more than one dex lower than that of the star formation main sequence), and covering more than an order of magnitude in stellar mass. This sample allowed us to revisit the sample from the different selections, finding ∼90% overlap between these criteria. The total sample of 136 quiescent galaxies constructed in this study shows that those at z ∼ 3 − 5, including the DeepDive targets, typically exhibit weaker 4000 Å breaks and bluer colors than their lower-redshift counterparts, indicating generally younger stellar populations. Stacked spectra of sources grouped by the D n 4000 index reveal faint iron and magnesium absorption line features in the stellar continuum even for the low D n 4000 (D n 4000 < 1.35) subsample at high redshift (z ∼ 3). In addition, higher D n 4000 subsamples show fainter nebular emission lines. These results demonstrate that medium-resolution NIRSpec spectroscopy is essential for robustly characterizing the diversity and evolution of early quiescent galaxies. The large sample constructed in this paper will allow a statistical census of the properties of quiescent galaxies at high redshift to be obtained. All photometric and spectroscopic data from this study have been made publicly available.