We measured typical quasi-one-dimensional organic metals (Bechgaard salts), using high resolution angle-resolved photoemission spectroscopy. We observe neither quasiparticle features near the chemical potential nor traces of a Fermi surface. The spectra, indicating a deep pseudogap around the Fermi energy, are incompatible with a Fermi liquid picture. None of the alternative interpretations we explored can fully explain the experimental data.