X-ray absorption lines of highly ionized species such as O VII at about zero redshift have been firmly detected in the spectra of several active galactic nuclei. However, the location of the absorbing gas remains a subject of debate. To separate the Galactic and extragalactic contributions to the absorption, we have obtained Chandra LETG-HRC and Far Ultraviolet Spectroscopic Explorer observations of the black hole X-ray binary LMC X-3. We clearly detect the O VII Kα absorption line with an equivalent width of 20(14, 26) mÅ (90% confidence range). The Ne IX Kα absorption line is also detected, albeit marginally. A joint analysis of these lines, together with the nondetection of the O VII Kβ and O VIII Kα lines, gives the temperature, velocity dispersion, and hot oxygen column density as 1.3(0.7,1.8) × 10⁶ K, 79(62,132) km s^-1, and 1.9(1.2, 3.2) ×10¹⁶ cm^-2, assuming a collisional ionization equilibrium of the X-ray-absorbing gas and a Galactic interstellar Ne/O number ratio of 0.18. The X-ray data allow us to place a 95% confidence lower limit to the Ne/O ratio as 0.14, but the upper limit is not meaningfully constrained. The O VII line centroid and its relative shift from the Galactic O I Kα absorption line, detected in the same observations, are inconsistent with the systemic velocity of LMC X-3 (+310 km s^-1). The far-UV spectrum shows O VI absorption at Galactic velocities, but no O VI absorption is detected at the LMC velocity at greater than 3 σ significance. The measured Galactic O VI column density is higher than the value predicted from the O VII-bearing gas, indicating multiphase absorption. Both the nonthermal broadening and the decreasing scale height with the increasing ionization state further suggest an origin of the highly ionized gas in a supernova-driven galactic fountain. In addition, we estimate the warm and hot electron column densities from our detected O II Kα line in the LMC X-3 X-ray spectra and from the dispersion measure of a pulsar in the LMC vicinity. We then infer the O/H ratio of the gas to be 8 × 10^-5, consistent with the chemically enriched galactic fountain scenario. We conclude that the Galactic hot interstellar medium should in general substantially contribute to zero-redshift X-ray absorption lines in extragalactic sources.