GHRS and FOS ultraviolet spectra of the bright QSO H1821 + 643 (m(v) = 14.2, z(em) = 0.297) reveal the presence of strong O VI lambda lambda 1031.93, 1037.62 absorption systems at z(abs) = 0.225 and 0.297, the latter being at the redshift of the QSO itself. Ground-based galaxy redshift measurements by us and others reveal two emission-line galaxies near the redshift of the intervening system at z(abs) = 0.225, suggesting the existence of a galaxy group at this redshift. The intervening O VI absorption system is also detected in H I but is not detected in the lines of Si II, Si IV, C IV, or N V. These ionization characteristics can be explained by a low-density, extended (similar to 300 kpc) diffuse gas distribution that is photoionized by the metagalactic UV background if the gas has a metallicity of similar to 0.1 times solar. Such a photoionized gas may be associated with the extended halo of the luminous intervening spiral galaxy at a projected distance of 100 h(75)(-1) kpc, or with an intragroup medium. Alternatively, the absorption may be produced in hot collisionally ionized halo gas or in a hot intragroup medium. The associated system with z(abs) = 0.297 contains narrow and broad O VI absorption. The narrow absorption, which is also detected in H I, C III, C IV, and Si IV, can be modeled as gas photoionized by H1821 + 643 with roughly solar abundances. This gas is probably situated close to H1821 + 643. The broad O VI absorption that is centered at the emission redshift of H1821 + 643 may represent a weak and narrow example of the broad absorption line phenomena. Another possibility is that the broad O VI absorption occurs in 10(5)-10(6) K gas associated with a cooling flow in the rich, X-ray-luminous cluster surrounding H1821 + 643 or with a cooling flow in the host elliptical galaxy that H1821 + 643 resides in. However, the observed O vr column density is 60-120 times smaller than expected for gas in a simple cooling how passing through the 10(6)-10(5) K temperature regime. The strong radiative flux from H1821 + 643 could substantially modify the ionization of the cooling gas in the flow and may help explain the discrepancy.