We present Far Ultraviolet Spectroscopic Explorer (FUSE) observations of the O VI λλ1031.926, 1037.617 absorption lines associated with gas in and near the Milky Way, as detected in the spectra of a sample of 100 extragalactic targets and two distant halo stars. We combine data from several FUSE Science Team programs with guest observer data that were public before 2002 May 1. The sight lines cover most of the sky above Galactic latitude |b|>25deg-at lower latitude the ultraviolet extinction is usually too large for extragalactic observations. We describe the details of the calibration, alignment in velocity, continuum fitting, and manner in which several contaminants were removed-Galactic H2, absorption intrinsic to the background target and intergalactic Lyβ lines. This decontamination was done very carefully, and in several sight lines very subtle problems were found. We searched for O VI absorption in the velocity range -1200 to 1200 km s-1. With a few exceptions, we only find O VI in the velocity range -400 to 400 km s-1 the exceptions may be intergalactic O VI. In this paper we analyze the O VI associated with the Milky Way (and possibly with the Local Group). We discuss the separation of the observed O VI absorption into components associated with the Milky Way halo and components at high velocity, which are probably located in the neighborhood of the Milky Way. We describe the measurements of equivalent width and column density, and we analyze the different contributions to the errors. We conclude that low-velocity Galactic O VI absorption occurs along all sight lines-the few nondetections only occur in noisy spectra. We further show that high-velocity O VI is very common, having equivalent width >65 mÅ in 50% of the sight lines and equivalent width >30 mÅ in 70% of the high-quality sight lines. The central velocities of high-velocity O VI components range from |vLSR|=100 to 330 km s-1 there is no correlation between velocity and absorption strength. We discuss the possibilities for studying O VI absorption associated with Local Group galaxies and conclude that O VI is probably detected in M31 and M33. We limit the extent of an O VI halo around M33 to be <100 kpc [at a 3 σ detection limit of log N(O VI)~14.0]. Using the measured column densities, we present 50 km s-1 wide O VI channel maps. These show evidence for the imprint of Galactic rotation. They also highlight two known H I high-velocity clouds (complex C and the Magellanic Stream). The channel maps further show that O VI at velocities <-200 km s-1 occurs along all sight lines in the region l=20deg-150deg, b<-30deg, while O VI at velocities >200 km s-1 occurs along all sight lines in the region l=180deg-300deg, b>20deg.