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 $mathcal{MACT}$, which will obtain rest-frame optical spectra for $sim$3000 emission-line galaxies. This paper describes the optical spectroscopy that has been conducted with MMT/Hectospec and Keck/DEIMOS for $approx$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 $mathcal{MACT}$ survey (Ly et al.) presents the first results on the stellar mass--gas metallicity relation at $zlesssim1$ using the sample with [OIII]$lambda$4363 measurements.
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