We have conducted a survey of near-contact binary systems observed during the ROSAT All-Sky Survey (RASS). The near-contact binaries (NCBs) have an A- or F-type primary, with a companion which is one to two spectral types cooler, The systems have periods less than 1 day and display strong tidal interaction, but they are not in contact like the W UMa systems, There are more than 150 such systems known to exist. We have analyzed the RASS data for all (58) of those known to lie within 400 pc. We report the detection of 14 systems with X-ray count rates >0.01 counts s-1. The X-ray luminosity function for the NCBs derived from this sample is very similar to that for A-type W UMa systems (derived, admittedly, from only a handful of Einstein observations) but appears to be significantly different from those of W-type W UMa systems and RS CVn binaries. This is consistent with the proposed scenario that the NCBs are evolutionary precursors to the A-type W UMa binaries. The mean X-ray luminosity of the NCBs is log LX = 29.3±0.1 ergs s-1, less than that of the RS CVns, but greater than that of normal late-type main-sequence stars. The LX/Lbol values for the handful of stars for which bolometric luminosities could be determined are consistent with their being near saturation. The detection of these systems may help to explain why many presumably single A-type stars were detected in the RASS; i.e., the "single" A stars may, in fact, be binaries, like the NCBs, with late-type companions.