Using the Eris zoom-in cosmological simulation of assembly of a Milky Way analogue, we study the chemical enrichment of stars due to accretion of metal-enriched gas from the interstellar medium (ISM) during the Galaxy’s development. We consider metal-poor and old stars in the Galactic halo and bulge through the use of stellar orbits, gas density and metallicity distributions in Eris. Assuming spherically symmetric Bondi–Hoyle accretion, we find that halo and bulge stars accrete metals at the rate of about 10$\mathrm{<mrow\; data-mjx-texclass="ORD">-24</mrow>}$ and 10$\mathrm{<mrow\; data-mjx-texclass="ORD">-22</mrow>}$ M$\odot$ yr$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$, respectively, at redshifts $z\lesssim$3,butthisaccretionrateincreasesroughlyahundred-foldtoabout10$^{−20}$ M$\odot$ yr$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ at higher redshifts due to increased gas density. Bulge and halo stars accrete similar amounts of metals at high redshifts when kinematically distinct bulge and halo have not yet developed, and both sets of stars encounter a similar metal distribution in the ISM. Accretion alone can enrich main-sequence stars up to [Fe/H] ~ −2 in extreme cases, with the median enrichment level due to accretion of about [Fe/H] ~ −6 to −5. Because accretion mostly takes place at high redshifts, it is $\alpha$-enriched to [$\alpha$/Fe] ~ 0.5. We find that accretive metal enrichment is sufficient to affect the predicted metallicity distribution function of halo stars at [Fe/H] < −5. This can hinder attempts to infer natal chemical environment of metal-poor stars from their observed enrichment. Peculiar enrichment patterns such as those predicted to arise from pair-instability supernovae could help in disentangling the natal and accreted metal content of stars.