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The radio source 3C 264, hosted by the giant elliptical galaxy NGC 3862, was observed with VERITAS between February 2017 and May 2019. These deep observations resulted in the discovery of very-high-energy (VHE; E $>100$ GeV) $γ$-ray emission from this active galaxy. An analysis of $\sim$57 hours of quality-selected live time yields a detection at the position of the source, corresponding to a statistical significance of 7.8 standard deviations above background. The observed VHE flux is variable on monthly time scales, with an elevated flux seen in 2018 observations. The VHE emission during this elevated state is well-characterized by a power-law spectrum with a photon index $Γ= 2.20 \pm 0.27$ and flux F($>315$ GeV) = ($7.6\pm 1.2_{\mathrm stat} \pm 2.3_{\mathrm syst})\times 10^{-13}$ cm$^{-2}$ s$^{-1}$, or approximately 0.7% of the Crab Nebula flux above the same threshold. 3C 264 ($z = 0.0217$) is the most distant radio galaxy detected at VHE, and the elevated state is thought to be similar to that of the famously outbursting jet in M 87. Consequently, extensive contemporaneous multi-wavelength data were acquired in 2018 at the time of the VHE high state. An analysis of these data, including VLBA, VLA, HST, Chandra and Swift observations in addition to the VERITAS data, is presented, along with a discussion of the resulting spectral energy distribution.