We report the discovery of an eclipsing X-ray binary with a 3.62-hr period within 24 of the center of the dwarf starburst galaxy NGC 4214. The orbital period places interesting constraints on the nature of the binary, and allows for a few very different interpretations. The most likely possibility is that the source lies within NGC 4214 and has an X-ray luminosity of up to 7 e38 ergs/s. In this case the binary may well be comprised of a naked He-burning donor star with a neutron-star accretor, though a stellar-mass black-hole accretor cannot be completely excluded. There is no obvious evidence for a strong stellar wind in the X-ray orbital light curve that would be expected from a massive He star; thus, the mass of the He star should be <3-4 solar masses. If correct, this would represent a new class of very luminous X-ray binary -- perhaps related to Cyg X-3. Other less likely possibilities include a conventional low-mass X-ray binary that somehow manages to produce such a high X-ray luminosity and is apparently persistent over an interval of years; or a foreground AM Her binary of much lower luminosity that fortuitously lies in the direction of NGC 4214. Any model for this system must accommodate the lack of an optical counterpart down to a limiting magnitude of 22.6 in the visible.
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