The first detected interstellar object Oumuamua that passed within 0.25au of the Sun on 2017 September 9 was presumably ejected from a stellar system. We use its newly determined non-Keplerian trajectory together with the reconstructed Galactic orbits of 7 million stars from Gaia DR2 to identify past close encounters. Such an encounter could reveal the home system from which Oumuamua was ejected. The closest encounter, at 0.60pc (0.53-0.67pc, 90% confidence interval), was with the M2.5 dwarf HIP 3757 at a relative velocity of 24.7km/s, 1Myr ago. A more distant encounter (1.6pc) but with a lower encounter (ejection) velocity of 10.7km/s was with the G5 dwarf HD 292249, 3.8Myr ago. Two more stars have encounter distances and velocities intermediate to these. The encounter parameters are similar across six different non-gravitational trajectories for Oumuamua. Ejection of Oumuamua by scattering from a giant planet in one of the systems is plausible, but requires a rather unlikely configuration to achieve the high velocities found. A binary star system is more likely to produce the observed velocities. None of the four home candidates have published exoplanets or are known to be binaries. Given that the 7 million stars in Gaia DR2 with 6D phase space information is just a small fraction of all stars for which we can eventually reconstruct orbits, it is a priori unlikely that our current search would find Oumuamuas home star system. As Oumuamua is expected to pass within 1pc of about 20 stars and brown dwarfs every Myr, the plausibility of a home system depends also on an appropriate (low) encounter velocity.
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