Carbon nanothreads are a new one-dimensional sp3-bonded nanomaterial of CH stoichiometry synthesized from benzene at high pressure and room temperature by slow solid-state polymerization. The resulting threads assume crystalline packing hundreds of micrometers across. We show high-resolution electron microscopy (HREM) images of hexagonal arrays of well-aligned thread columns that traverse the 80-100 nm thickness of the prepared sample. Diffuse scattering in electron diffraction reveals that nanothreads are packed with axial and/or azimuthal disregistry between them. Layer lines in diffraction from annealed nanothreads provide the first evidence of translational order along their length, indicating that this solid-state reaction proceeds with some regularity. HREM also reveals bends and defects in nanothread crystals that can contribute to the broadening of their diffraction spots, and electron energy-loss spectroscopy confirms them to be primarily sp3-hybridized, with less than 27% sp2 carbon, most likely associated with partially saturated "degree-4" threads.