  Gaia BH3 is the first observed dormant black hole (BH) with a mass of
$\approx{30}$ M$_\odot$ and represents the first confirmation that such massive
BHs are associated with metal-poor stars. Here, we explore the isolated binary
formation channel for Gaia BH3 focusing on the old and metal-poor stellar
population of the Milky Way halo. We use the MIST stellar models and our
open-source population synthesis code SEVN to evolve $5.6 \times 10^8$ binaries
exploring 20 sets of parameters. We find that systems like Gaia BH3 form
preferentially from binaries initially composed of a massive star ($40-60$
M$_\odot$) and a low mass companion ($<1$ M$_\odot$) in a wide ($P>10^3$ days)
and eccentric orbit ($e>0.6$). Such progenitor binary stars do not undergo any
Roche-lobe overflow episode during their entire evolution, so that the final
orbital properties of the BH-star system are determined at the core collapse of
the primary star. Low natal kicks ($\lesssim$ 10~km/s) significantly favour the
formation of Gaia BH3-like systems, but high velocity kicks up to $\approx 220$
km/s are also allowed. We estimate the formation efficiency for Gaia BH3-like
systems in old ($t>10$ Gyr) and metal-poor ($Z<0.01$) populations to be $\sim 4
\times 10^{-8}$ M$_\odot^{-1}$ (for our fiducial model), representing $\sim
3\%$ of the whole simulated BH-star population. We expect up to $\approx 4000$
BH-star systems in the Galactic halo formed through isolated evolution, of
which $\approx 100$ are compatible with Gaia BH3-like. Given the density
profile of the Galactic halo we do not expect more than one at the observed
distance of Gaia BH3. Our models show that, even if it was born inside a
stellar cluster, Gaia BH3 is compatible with a primordial binary star that
escaped from its parent cluster without experiencing significant dynamical
interactions.
