We present ASCA observations of low-ionization nuclear emission-line regions (LINERs) without broad Hα emission in their optical spectra. The sample of ``type 2'' LINERs consists of NGC 404, 4111, 4192, 4457, and 4569. We have detected X-ray emission from all the objects except for NGC 404; among the detected objects are two so-called transition objects (NGC 4192 and NGC 4569), which have been postulated to be composite nuclei having both an H II region and a LINER component. The images of NGC 4111 and NGC 4569 in the soft (0.5-2 keV) and hard (2-7 keV) X-ray bands are extended on scales of several kiloparsecs. The X-ray spectra of NGC 4111, NGC 4457, and NGC 4569 are well fitted by a two-component model that consists of soft thermal emission with kT~0.65 keV and a hard component represented by a power law (photon index ~2) or by thermal bremsstrahlung emission (kT~several keV). The extended hard X-rays probably come from discrete sources, while the soft emission most likely originates from hot gas produced by active star formation in the host galaxy. We have found no clear evidence for the presence of active galactic nuclei (AGNs) in the sample. Using black hole masses estimated from host galaxy bulge luminosities, we obtain an upper limit on the implied Eddington ratios less than 5×10-5. If an AGN component is the primary ionization source of the optical emission lines, then it must be heavily obscured with a column density significantly larger than 1023 cm-2, since the observed X-ray luminosity is insufficient to drive the luminosities of the optical emission lines. Alternatively, the optical emission could be ionized by a population of exceptionally hot stars. This interpretation is consistent with the small [O I] λ6300/Hα ratios observed in these sources, the ultraviolet spectral characteristics in the cases where such information exists, and the X-ray results reported here. We also analyze the X-ray properties of NGC 4117, a low-luminosity Seyfert 2 galaxy serendipitously observed in the field of NGC 4111.