Weak spectral features in BL Lacertae objects (BL Lacs) often provide a unique opportunity to probe the inner region of this rare type of active galactic nucleus. We present a Hubble Space Telescope/Cosmic Origins Spectrograph observation of the BL Lac H 2356-309. A weak Lyα emission line was detected. This is the fourth detection of a weak Lyα emission feature in the ultraviolet (UV) band in the so-called high-energy peaked BL Lacs, after Stocke et al. Assuming the line-emitting gas is located in the broad line region (BLR) and the ionizing source is the off-axis jet emission, we constrain the Lorentz factor (Γ) of the relativistic jet to be ≥8.1 with a maximum viewing angle of 3.°6. The derived Γ is somewhat larger than previous measurements of Γ ≈ 3-5, implying a covering factor of ∼3% of the line-emitting gas. Alternatively, the BLR clouds could be optically thin, in which case we constrain the BLR warm gas to be ∼10-5 M⊙. We also detected two H I and one O VI absorption lines that are within |Δv| < 150 km s-1 of the BL Lac object. The O VI and one of the H I absorbers likely coexist due to their nearly identical velocities. We discuss several ionization models and find a photoionization model where the ionizing photon source is the BL Lac object that can fit the observed ion column densities with reasonable physical parameters. This absorber can either be located in the interstellar medium of the host galaxy or in the BLR.