The edge-on spiral galaxy NGC 4565 was observed for similar to 35 ks with ASCA in the 0.5-10 keV energy band. The X-ray emission was dominated by two bright sources, which could be identified with two point-like X-ray sources seen in the ROSAT HRI image. The observed 0.5-10 keV fluxes of these sources, 1.7 x 10(-12) erg s(-1) cm(-2) and 0.7 x 10(-12) erg s(-1) cm(-2), imply bolometric luminosities of 1.2 x 10(40) erg s(-1) and 4.6 x 10(39) erg s(-1), respectively. They exhibit similar spectra, which can be explained by emission from optically thick accretion disks with inner-disk temperature of 1.4-1.6 keV. One of them, coincident in position with the nucleus, shows too low absorption to be the active nucleus seen through the galaxy disk. Their spectra and high luminosities suggest that they are both mass-accreting black-hole binaries. However, the black-hole mass required by the Eddington limit is rather high (greater than or equal to 50M.), and the observed disk temperature is too high to be compatible with the high black-hole mass. Several attempts have been made to solve these problems.