We report on the discovery of FRB 20200120E, a repeating fast radio burst (FRB) with low dispersion measure (DM), detected by the Canadian Hydrogen Intensity Mapping Experiment (CHIME)/FRB project. The source DM of 87.82 pc cm$^{-3}$ is the lowest recorded from an FRB to date, yet is significantly higher than the maximum expected from the Milky Way interstellar medium in this direction (~ 50 pc cm$^{-3}$). We have detected three bursts and one candidate burst from the source over the period 2020 January-November. The baseband voltage data for the event on 2020 January 20 enabled a sky localization of the source to within $simeq$ 14 sq. arcmin (90% confidence). The FRB localization is close to M81, a spiral galaxy at a distance of 3.6 Mpc. The FRB appears on the outskirts of M81 (projected offset $sim$ 20 kpc) but well inside its extended HI and thick disks. We empirically estimate the probability of chance coincidence with M81 to be $< 10^{-2}$. However, we cannot reject a Milky Way halo origin for the FRB. Within the FRB localization region, we find several interesting cataloged M81 sources and a radio point source detected in the Very Large Array Sky Survey (VLASS). We searched for prompt X-ray counterparts in Swift/BAT and Fermi/GBM data, and for two of the FRB 20200120E bursts, we rule out coincident SGR 1806$-$20-like X-ray bursts. Due to the proximity of FRB 20200120E, future follow-up for prompt multi-wavelength counterparts and sub-arcsecond localization could be constraining of proposed FRB models.
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