We present detailed analyses of the absorption spectrum seen in QSO 2359 - 1241 (NVSS J235953 - 124148). Keck HIRES data reveal absorption from 20 transitions arising from He I, Mg I, Mg II, Ca II, and Fe II. Hubble Space Telescope data show broad absorption lines (BALs) from Al III lambda 1857, C IV lambda 1549, Si IV lambda 1397, and N V lambda 1240. Absorption from excited Fe II states constrains the temperature of the absorber to 2000 less than or similar to T less than or similar to 10,000 K and puts a lower limit of 10(5) cm(-3) on the electron number density. Saturation diagnostics show that the real column densities of He I and Fe II can be determined, allowing us to derive meaningful constraints on the ionization equilibrium and abundances in the flow. The ionization parameter is constrained by the iron, helium, and magnesium data to -3.0 less than or similar to log (U) less than or similar to -2.5, and the observed column densities can be reproduced without assuming departure from solar abundances. From comparison of the He I and Fe II absorption features, we infer that the outflow seen in QSO 2359 - 1241 is not shielded by a hydrogen ionization front and therefore that the existence of low-ionization species in the outflow (e.g., Mg II, Al III, Fe II) does not necessitate the existence of such a front. We find that the velocity width of the absorption systematically increases as a function of ionization and to a lesser extent with abundance. Complementary analyses of the radio and polarization properties of the object are discussed in a companion paper (Brotherton et al.).