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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.