We use high-resolution UV spectra taken by the Space Telescope Imaging Spectrograph instrument on board the Hubble Space Telescope to study the 5.5 pc line of sight to the K0 V star 36 Oph A. The one interstellar component detected for this line of sight has a velocity inconsistent with the local interstellar cloud (LIC) flow vector but consistent with the flow vector of the so-called G cloud, a very nearby warm cloud in the Galactic center direction. From our data, we measure the following values for the interstellar temperature, nonthermal velocity, H I column density, and D/H value: T = 5900 ± 500 K, ξ = 2.2 ± 0.2 km s-1, log NH = 17.85 ± 0.15, and D/H = (1.5 ± 0.5) × 10-5. The temperature of the G cloud is somewhat lower than that of the LIC, and Mg and Fe depletions are also lower, but the D/H value appears to be the same. Based on upper limits for the LIC absorption, we estimate the distance to the edge of the LIC to be dedge < 0.19 pc, which the Sun will reach in tedge < 7400 yr. The H I Lyα absorption line has properties inconsistent with those of the other absorption lines, indicating the presence of one or more absorption components not seen in the other lines. We present evidence that hot neutral hydrogen local to both the Sun and the star is responsible for the excess Lyα absorption. This hot H I is created by the interaction between the ISM and the winds of the Sun and 36 Oph A. The observed line of sight lies only 12° from the upwind direction of the LIC flow vector, where hydrodynamic models of the heliosphere suggest that heliospheric H I absorption should be particularly prominent. The properties of the heliospheric absorption (T = 38,000 ± 8000 K and log NH = 14.6 ± 0.3) are consistent with previous measurements of this absorption for the α Cen line of sight 52° from the upwind direction.