Far-UV echelle spectroscopy of the radio-quiet QSO H1821+643 (zem=0.297), obtained with the Space Telescope Imaging Spectrograph (STIS) at approximately 7 km s-1 resolution, reveals four definite O vi absorption-line systems and one probable O vi absorber at 0.15<zabs<0.27. The four definite O vi absorbers are located near galaxies and are highly displaced from the quasar in redshift; these are likely intervening systems unrelated to the background QSO. In the case of the strong O vi system at zabs=0.22497, multiple components are detected in Si iii and O vi as well as H i Lyman series lines, and the differing component velocity centroids and b-values firmly establish that this is a multiphase absorption system. A weak O vi absorber is detected at zabs=0.22637, i.e., offset by approximately 340 km s-1 from the zabs=0.22497 system. Lyalpha absorption is detected at zabs=0.22613, but no Lyalpha absorption is significantly detected at 0.22637. Other weak O vi absorbers at zabs=0.24531 and 0.26659 and the probable O vi system at 0.21326 have widely diverse O vi/H i column density ratios with N(O vi)/N(H i) ranging from </=0.14+/-0.03 to 5.2+/-1.2. The number density of O vi absorbers with rest equivalent width greater than 30 mÅ in the H1821+643 spectrum is remarkably high, dN&solm0;dz approximately 48, which implies with a high (90%) confidence that it is greater than 17 in the low-redshift intergalactic medium. We conservatively estimate that the cosmological mass density of the O vi systems is Omegab(Ovi&parr0; greater, similar0.0008 h-175. With an assumed metallicity of 1/10 solar and a conservative assumption that the fraction of oxygen in the O vi ionization stage is 0.2, we obtain Omegab(Ovi&parr0; greater, similar0.004 h-175. This is comparable to the combined cosmological mass density of stars and cool gas in galaxies and X-ray-emitting gas in galaxy clusters at low redshift.