  Gravitational-wave (GW) detections of electromagnetically bright compact
binary coalescences can provide an independent measurement of the Hubble
constant $H_0$. In order to obtain a measurement that could help arbitrating
the existing tension on $H_0$, one needs to fully understand any source of
systematic biases for this approach. In this study, we aim at understanding the
impact of instrumental calibration errors (CEs) and uncertainties on luminosity
distance measurements, $D_L$, and the inferred $H_0$ results. We simulate
binary neutron star mergers (BNSs), as detected by a network of Advanced LIGO
and Advanced Virgo interferometers at their design sensitivity. We artificially
add CEs equal to exceptionally large values experienced in LIGO-Virgo's third
observing run (O3). We find that for individual BNSs at a network
signal-to-noise ratio of 50, the systematic errors on $D_L$ - and hence $H_0$ -
are still smaller than the statistical uncertainties. The biases become more
significant when we combine multiple events to obtain a joint posterior on
$H_0$. In the rather unrealistic case that the data around each detection is
affected by the same CEs corresponding to the worst offender of O3, the true
$H_0$ value would be excluded from the 90% credible interval after $\sim40$
sources. If instead 10% of the sources suffer from severe CEs, the true value
of $H_0$ is included in the 90% credible interval even after we combine 100
sources.
