The discovery of a z(abs) = 0.0912 damped Ly alpha absorption-line system in the Hubble Space Telescope Faint Object Spectrograph ultraviolet spectrum of the quasar OI 363 (0738+313) is reported. This is the lowest redshift quasar-damped Ly alpha system known. Its neutral hydrogen column density is N(H I) = 1.5 +/- 0.2 x 10(21) atoms cm(-2), which easily exceeds the classical criterion for damped Ly alpha of N(H I) greater than or equal to 2 x 10(20) atoms cm(-2). Remarkably, a z(abs) = 0.2212 damped system with N(H I) = 7.9 +/- 1.4 x 10(20) atoms cm(-2) has also been discovered in the same spectrum. In the past, the standard paradigm for damped Ly alpha systems has been that they arise in galactic or protogalactic H I disks with low impact parameters in luminous galaxies. However WIYN imaging of the OI 363 field shows that none of the galaxies visible in the vicinity of the quasar is a luminous gas-rich spiral with low impact parameter, either at z = 0.0912 or z = 0.2212. Thus, these damped systems are among the dearest examples yet of cases that are inconsistent with the standard damped Ly alpha-H I-disk paradigm.