We have determined spectroscopic ages of elliptical galaxies in the Virgo Cluster using spectra of very high signal-to-noise ratio (S/N>100 Å-1). We observed eight galaxies with the Subaru Telescope and have combined this sample with six galaxies previously observed with the WHT. To determine their ages, we have used a new method based on the Hγσ age indicator, which is virtually independent of the effects of metallicity. Apart from ages we have estimated abundances of various elements. In this paper we present the observations, the data reduction, and the reliability of the Hγσ method. The results of this investigation are presented in a companion paper.