  We present a study of the cold atomic hydrogen (HI) content of molecular
clouds simulated within the SILCC-Zoom project. We produce synthetic
observations of HI at 21 cm including HI self-absorption (HISA) and
observational effects. We find that HI column densities, $N_\textrm{HI}$, of
$\gtrsim$10$^{22}$ cm$^{-2}$ are frequently reached in molecular clouds with HI
temperatures as low as $\sim$10 K. Hence, HISA observations assuming a fixed HI
temperature tend to underestimate the amount of cold HI in molecular clouds by
a factor of 3 - 10 and produce an artificial upper limit of $N_\textrm{HI}$
around 10$^{21}$ cm$^{-2}$. We thus argue that the cold HI mass in clouds could
be a factor of a few higher than previously estimated. Also
$N_\textrm{HI}$-PDFs obtained from HISA observations might be subject to
observational biases and should be considered with caution. The underestimation
of cold HI in HISA observations is due to both the large HI temperature
variations and the effect of noise in regions of high optical depth. We find
optical depths of cold HI around 1 - 10 making optical depth corrections
essential. We show that the high HI column densities ($\gtrsim$10$^{22}$
cm$^{-2}$) can in parts be attributed to the occurrence of up to 10 individual
HI-H$_2$ transitions along the line of sight. This is also reflected in the
spectra, necessitating Gaussian decomposition algorithms for their analysis.
However, also for a single HI-H$_2$ transition, $N_\textrm{HI}$ frequently
exceeds 10$^{21}$ cm$^{-2}$, challenging 1D, semi-analytical models. This is
due to non-equilibrium chemistry effects and the fact that HI-H$_2$ transition
regions usually do not possess a 1-dimensional geometry. Finally, we show that
the HI gas is moderately supersonic with Mach numbers of a few. The
corresponding non-thermal velocity dispersion can be determined via HISA
observations within a factor of $\sim$2.
