We simulate the passage through the Sun-Jupiter system of interstellar objects (ISOs) similar to 1I/`Oumuamua or 2I/Borisov. Capture of such objects is rare and overwhelmingly from low incoming speeds onto orbits akin to those of known long-period comets. This suggests that some of these comets could be of extra-solar origin, in particular inactive ones. Assuming ISOs follow the local stellar velocity distribution, we infer a volume capture rate of $0.051,mathrm{au}^3 mathrm{yr}^{-1}$. Current estimates for orbital lifetimes and space densities then imply steady-state captured populations of $sim10^2$ comets and $sim10^5$ `Oumuamua-like rocks, of which 0.033% are within 6au at any time.
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