We report results of Chandra X-ray and Very Large Array (VLA) radio observations of the Galactic accreting black hole V404 Cyg (GS 2023+338) in its quiescent state. V404 Cyg is detected at its faintest level of radio and X-ray emission with a 0.5–10 keV unabsorbed luminosity of 8.3 × 1032 (d/3.5 kpc)2 erg s−1. The X-ray spectrum fit with an absorbed power-law model yields a photon index of 2.17 ± 0.13. Contrary to previous findings, this clearly indicates that V404 Cyg undergoes – like most black holes in quiescence – a softening of its X-ray spectrum at very low luminosity compared to the standard hard state. The quiescent radio emission is consistent with the presence of self-absorbed compact jets. We have also re-analysed archival data from the decay of the 1989 outburst of V404 Cyg in order to quantify more precisely the correlation between radio and X-ray emission in the hard state of V404 Cyg. We show that this correlation extends over five decades in X-ray flux and holds down to the quiescent state of V404 Cyg. The index of this correlation (∼0.5) may suggest that synchrotron self-Compton emission is the dominant physical process at high energy in V404 Cyg. However, this index is also consistent with scale invariant jet models coupled to an inefficiently radiating accretion disc. We discuss the properties of the quiescent state of black holes and highlight the fact that some of their properties are different from the standard hard state.