Deep rest-frame optical spectroscopy is critical for characterizing and understanding the physical conditions and properties of the ionized gas in galaxies. Here, we present a new spectroscopic survey called "Metal Abundances across Cosmic Time" or MACT, which will obtain rest-frame optical spectra for ~3000 emission-line galaxies. This paper describes the optical spectroscopy that has been conducted with MMT/Hectospec and Keck/DEIMOS for ~1900 z=0.1-1 emission-line galaxies selected from our narrowband and intermediate-band imaging in the Subaru Deep Field. In addition, we present a sample of 164 galaxies for which we have measured the weak [OIII]{lambda}4363 line (66 with at least 3{sigma} detections and 98 with significant upper limits). This nebular emission line determines the gas-phase metallicity by measuring the electron temperature of the ionized gas. This paper presents the optical spectra, emission-line measurements, interstellar properties (e.g., metallicity, gas density), and stellar properties (e.g., star formation rates, stellar mass). Paper II of the MACT survey (Ly et al. 2016ApJ...828...67L) presents the first results on the stellar mass-gas metallicity relation at z<=1 using the sample with [OIII]{lambda}4363 measurements.