We present intermediate-resolution (FWHM similar to 10-20 km s(-1)) GHRS observations of interstellar and intergalactic absorption lines in the spectrum of the bright quasar H1821+643. The interstellar species detected include S II, Si II, Mg II, C IV, and N V. The intergalactic lines include Ly alpha (six lines), Ly beta (2 lines), Ly gamma, and C III. The Galactic direction of H1821+643 (l = 94.degrees 0 b = 27.degrees 4) allows for the study of gas in the warped outer Galaxy since the effects of Galactic rotation cause absorption by distant gas to be Doppler-shifted by velocities approaching -190 km s(-1). We detect by Mg II and Si II extending from approximately +40 km s(-1) to -150 km s(-1), indicating the presence of gas containing these metals in the outer Galaxy. Over the velocity range from - 100 to - 150 km s(-1), we find Si II/H I, Mg II/H I, and S II/H I = 0.92 +/- 0.20, greater than or equal to 0.25, and less than or equal to 1.5 times the abundances of these elements in the Sun. However, the presence of C IV in this gas implies corrections must be made for the effects of ionization. A simple model allowing for the ionization by the EUV background suggests the measurements are consistent with absorption in a gas with an intrinsic metallicity of roughly 1/10 solar. Strong C IV and N V absorption is found in broad absorption centered near -8 km s(-1). This absorption is well represented with a component having a Doppler spread parameter, b similar to 32 km s(-1), N(C IV) = (10.4 +/- 1.4) x 10(13) cm(-2), and N(N V) = (14.3 +/- 2.6) x 10(13) cm(-2). The inferred column density ratio of C IV to N V of approximately 0.7 and the large line width suggests this gas is probably hot (T > 2 x 10(5) K) and may be associated with Galactic radio loop III. The C IV doublet absorption also exhibits strong components centered near -70 and -120 km s(-1), which are not seen in N V. The -70 km s(-1) C IV absorption probably is associated with highly ionized gas situated 1.5 kpc above the Perseus spiral arm. The -120 km s(-1) absorption is likely associated with gas above the outer spiral arm. of the Milky Way. Assuming corotation of disk and halo gas, a velocity of -120 km s(-1) occurs at a Galactocentric distance of 24 kpc and a distance away of the plane of 7 kpc. The ionization of this distant gas may be produced by photoionization from the EUV extragalactic background radiation. We observe five intergalactic Ly alpha lines with W-lambda(rest) > 60 m Angstrom over a redshift path of 0.057 implying a low-redshift Ly alpha cloud density in agreement with the local number density found toward 3C 273 by Morris et al. (1991). Combining the results for the two sight lines we obtain a Ly alpha cloud density dN/dz = 73 +/- 19 for 0 < z < 0.3.