  Astrophysical models of binary-black hole mergers in the Universe require a
significant fraction of stellar-mass black holes (BHs) to receive negligible
natal kicks to explain the gravitational wave detections. This implies that BHs
should be retained even in open clusters with low escape velocities
($\lesssim1~\mathrm{km \, s^{-1}}$). We search for signatures of the presence
of BHs in the nearest open cluster to the Sun - the Hyades - by comparing
density profiles of direct $N$-body models to data from $Gaia$. The
observations are best reproduced by models with $2-3$ BHs at present. Models
that never possessed BHs have an half-mass radius $\sim30\%$ smaller than the
observed value, while those where the last BHs were ejected recently
($\lesssim150~$Myr ago) can still reproduce the density profile. In 50% of the
models hosting BHs, we find BHs with stellar companion(s). Their period
distribution peaks at $\sim10^3$ yr, making them unlikely to be found through
velocity variations. We look for potential BH companions through large $Gaia$
astrometric and spectroscopic errors, identifying 56 binary candidates - none
of which consistent with a massive compact companion. Models with $2-3$ BHs
have an elevated central velocity dispersion, but observations can not yet
discriminate. We conclude that the present-day structure of the Hyades requires
a significant fraction of BHs to receive natal kicks smaller than the escape
velocity of $\sim 3\, \mathrm{km \, s^{-1}}$ at the time of BH formation and
that the nearest BHs to the Sun are in, or near, Hyades.
