We present ASCA results on four prototype powerful far-infrared galaxies, Mrk 231, Mrk 273, Arp 220 and NGC 6240. The soft X-ray spectra show signatures of thermal emission with temperatures of (0.5-1)x10(7) K, which is probably produced in starbursts. Their soft X-ray (0.5-2 keV) luminosities range from 4x10(40) to 7 x 10(41) erg s(-1). The X-ray properties are examined in the context of a starburst. Evidence for a heavily obscured active nucleus is found in Mrk 273 and NGC 6240. The ASCA spectra of both galaxies show strong iron K emission-line features. The hard X-ray emission (>3 keV) in NGC 6240 is most likely the reflected light of a hidden QSO, the intrinsic luminosity of which is suspected to be similar to 10(45) erg s(-1) comparable to the far-IR luminosity. The 2-10 keV emission, possibly related to an active galactic nucleus (AGN), is found in Mrk 231. The observed 2-10 keV luminosity is only similar to 2 x 10(42) erg s(-1), and the origin of the hard X-ray emission is uncertain because of the low quality of the present data. No evidence for an AGN is found in Arp 220 in the X-ray data. However, the soft X-ray emission originating in the starburst is also as weak as the Her nebula and near-IR continuum, despite the large far-IR excess. The possible existence of a powerful but heavily obscured (Compton-thick) AGN is discussed.