Since fast radio bursts (FRBs) were discovered, their precise origins have remained a mystery. Multiwavelength observations of nearby FRB sources provide one of the best ways to make rapid progress in our understanding of the enigmatic FRB phenomenon. We present results from a sensitive, broadband multiwavelength X-ray and radio observational campaign of FRB 20200120E, the closest known extragalactic repeating FRB source. At a distance of 3.63 Mpc, FRB 20200120E resides in an exceptional location, within a ~10 Gyr-old globular cluster in the M81 galactic system. We place deep limits on both the persistent X-ray luminosity and prompt X-ray emission at the time of radio bursts from FRB 20200120E, which we use to constrain possible progenitors for the source. We compare our results to various classes of X-ray sources and transients. In particular, we find that FRB 20200120E is unlikely to be associated with: ultraluminous X-ray bursts (ULXBs), similar to those observed from objects of unknown origin in other extragalactic globular clusters; giant flares, like those observed from Galactic and extragalactic magnetars; or most intermediate flares and very bright short X-ray bursts, similar to those seen from magnetars in the Milky Way. We show that FRB 20200120E is also unlikely to be powered by a persistent or transient ultraluminous X-ray (ULX) source or a young, extragalactic pulsar embedded in a Crab-like nebula. We also provide new constraints on the compatibility of FRB 20200120E with accretion-based FRB models involving X-ray binaries and models that require a synchrotron maser process from relativistic shocks to generate FRB emission. These results highlight the power that multiwavelength observations of nearby FRBs can provide for discriminating between potential FRB progenitor models.