We report on X-ray observations of two soft X-ray transients containing neutron stars, 4U 2129+47 and EXO 0748-676. Our emphasis is on the quiescent properties of these sources. The X-ray spectrum and light curve of the eclipsing low-mass X-ray binary (LMXB) 4U 2129+47 was measured with the ROSAT Position Sensitive Proportinal Counter during its current quiescent state. The quiescent X-ray luminosity of ∼1032.7 ergs s-1 and blackbody temperature kT ∼ 0.21 keV are similar to other quiescent LMXBs. The quiescent X-ray light curve may show orbital modulation, but the statistics are insufficient to distinguish between a V-shaped partial eclipse (as seen in the high state) or a total square-wave eclipse. The similarity in the luminosity and temperature to other (noneclipsing) quiescent LMXBs implies that the vertical structure in the disk that blocked our direct view of the neutron star in the high state has collapsed, and the neutron star is seen directly. EXO 0748-676 was serendipitously observed with the Einstein IPC in quiescence before it was discovered as a bright transient with EXOSAT. Our reanalysis of this quiescent observation finds Lx ∼ 1034.0 ergs s-1, and blackbody temperature kT ∼ 0.22 keV, again similar to other LMXBs in quiescence.