  We report on the results of an XMM-Newton observation of the Supergiant Fast
X-ray Transient (SFXT) IGR J08408-4503 performed in June 2020. The source is
composed by a compact object (likely a neutron star) orbiting around an
O8.5Ib-II(f)p star, LM Vel. The X-ray light curve shows a very low level of
emission, punctuated by a single, faint flare. Analysis of spectra measured
during the flare and during quiescence is performed. The quiescent state shows
a continuum spectrum well deconvolved to three spectral models: two components
are from a collisionally-ionized plasma (with temperatures kT1=0.24 keV and
kT2=0.76 keV), together with a power law model (photon index of 2.55),
dominating above 2 keV. The X-ray flux emitted at this lowest level is
3.2$\times10^{-13}$ erg/cm2/s (0.5-10 keV, corrected for the interstellar
absorption), implying an X-ray luminosity of 1.85$\times10^{32}$ erg/s (at 2.2
kpc). The two temperature collisionally-ionized plasma is intrinsic to the
stellar wind of the donor star, while the power law can be interpreted as
emission due to residual, low level accretion onto the compact object. The
X-ray luminosity contributed by the power law component only, in the lowest
state, is (4.8$\pm{1.4})\times10^{31}$ erg/s, the lowest quiescent luminosity
detected from the compact object in an SFXT. Thanks to this very faint X-ray
state caught by XMM-Newton, X-ray emission from the wind of the donor star LM
Vel could be well-established and studied in detail for the first time, as well
as a very low level of accretion onto the compact object. The residual
accretion rate onto the compact object in IGR J08408-4503 can be interpreted as
the Bohm diffusion of (possibly magnetized) plasma entering the neutron star
magnetosphere at low Bondi capture rates from the supergiant donor wind at the
quasi-spherical radiation-driven settling accretion stage.
