We report the results of Hubble Space Telescope (HST) observations of the nearby (d = 5.0 pc) K1 V star 40 Eri A, which we use to search for scattered Lyα emission surrounding the star indicative of the interaction between the stellar wind and the interstellar medium (ISM). Absorption from circumstellar hot H I has previously been detected around many solar-like stars in HST observations of their Lyα lines, so there is potential for circumstellar Lyα emission to be detectable as well. There was previously a tentative detection of absorption for 40 Eri A, but unfortunately, we do not detect any circumstellar emission around 40 Eri A in our new observations. We use hydrodynamic models of the stellar "astrosphere" (i.e., the ISM interaction region) and radiative transfer calculations to demonstrate that emission should have been detected for assumed mass-loss rates of 2 ☉, assuming that the star is surrounded by warm, partially neutral ISM material like that which surrounds the Sun. In contrast, when the models are compared with the absorption data, we find consistency with the data only for 2 ☉. We believe that the most likely explanation for these apparently contradictory results is that the previous tentative detection of astrospheric absorption toward 40 Eri A is erroneous and that 40 Eri A probably lies within the hot ionized phase of the ISM. Thus, there is no interstellar H I within the astrosphere for us to detect in either absorption or emission, and no meaningful constraints on the mass-loss rate of 40 Eri A can be derived from the HST data.