From the first two seasons of the Palomar Transient Factory, we identify three peculiar transients (PTF 09dav, PTF 10iuv, and PTF 11bij) with five distinguishing characteristics: peak luminosity in the gap between novae and supernovae (M$_R$ ≈ -15.5 to -16.5 mag), rapid photometric evolution (t $_rise$ ≈ 12-15 days), large photospheric velocities (≈6000-11,000 km s$^-1$), early spectroscopic evolution into nebular phase (≈1-3 months), and peculiar nebular spectra dominated by calcium. We also culled the extensive decade-long Lick Observatory Supernova Search database and identified an additional member of this group, SN 2007ke. Our choice of photometric and spectroscopic properties was motivated by SN 2005E (Perets et al.). To our surprise, as in the case of SN 2005E, all four members of this group are also clearly offset from the bulk of their host galaxy. Given the well-sampled early- and late-time light curves, we derive ejecta masses in the range of 0.4-0.7 M $_☉$. Spectroscopically, we find that there may be a diversity in the photospheric phase, but the commonality is in the unusual nebular spectra. Our extensive follow-up observations rule out standard thermonuclear and standard core-collapse explosions for this class of ``calcium- rich gap’’ transients. If the progenitor is a white dwarf, we are likely seeing a detonation of the white dwarf core and perhaps even shock-front interaction with a previously ejected nova shell. If the progenitor is a massive star, a nonstandard channel specific to a low-metallicity environment needs to be invoked (e.g., ejecta fallback leading to black hole formation). Detection (or the lack thereof) of a faint underlying host (dwarf galaxy and cluster) will provide a crucial and decisive diagnostic to choose between these alternatives.