We present broadband and narrowband Hubble Space Telescope (HST) images of the central region of the heavily disturbed Virgo cluster galaxy NGC 4438 (Arp 120), whose nucleus has been described as a type 1 LINER or dwarf Seyfert galaxy. Narrowband Hα and [N II] HST images reveal striking bipolar shell features, 1 kpc in projected length from end to end, that are likely the result of an outflow from the nuclear region experiencing a strong interaction with the interstellar medium (ISM). While these outflow shells share similarities with those in some other starburst or active galactic nucleus (AGN) galaxies, those in NGC 4438 are notable because NGC 4438 harbors neither a luminous circumnuclear starburst nor a luminous AGN. The shells appear to be closed at their outer ends, suggesting that the outflow in NGC 4438 is dynamically younger than those in some other galaxies. The radio continuum emission is strongly enhanced near the outer ends of the shells, suggesting working surfaces arising from collimated nuclear outflows that have impacted and shocked the surrounding ISM. The two shells are quite different, as the northwestern shell is luminous and compact, while the southeastern shell is 2.5 times longer and much fainter in both optical emission lines and the nonthermal radio continuum. The differences between the two shells may be attributed to a difference in ISM density on the two sides of the nuclear disk. Such an ISM asymmetry exists on larger scales in this heavily disturbed galaxy. At the base of the outflow is a nuclear source, which is the highest surface brightness source in the galaxy at optical wavelengths. This source is resolved with a FWHM = 03 = 25 pc and has modest luminosities, uncorrected for extinction, of 5 × 1038 ergs s-1 in Hα and MB = -13. We discuss whether the outflow is powered by a low-luminosity AGN or a compact nuclear starburst. The kinetic energy associated with the ionized gas in the shells is ~1053 ergs, which could be produced by either massive star formation or an AGN. While the northwestern shell, which contributes most of the flux in most ground-based apertures centered on the nucleus, exhibits LINER-type line ratios, the nucleus has an Hα/[N II] ratio consistent with an H II region. Although there appears to be very little massive star formation occurring in the central kpc, the nucleus may contain a partially obscured, young nuclear star cluster. On the other hand, the collimation of the southeastern shell and the strongly enhanced radio continuum emission at the outer ends of the shells are more easily explained by jets associated with a nuclear black hole than by a compact nuclear starburst, although there is no direct evidence of a jet. The 2000 km s-1 broad-line component could be due to an AGN broad-line region but might also be due to high-velocity gas in the outflow. Since NGC 4438 has a large bulge, a large nuclear black hole might be expected.