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We measure abundances of 12 elements (Na, Mg, Si, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni) in a sample of 86 metal-poor ($-2 \lesssim \text{[Fe/H]} \lesssim -1$) subgiant stars in the solar neighborhood. Abundances are derived from high-resolution spectra taken with the Potsdam Echelle Polarimetric and Spectroscopic Instrument on the Large Binocular Telescope, modeled using iSpec and MOOG. By carefully quantifying the impact of photon-noise ($< 0.05$ dex for all elements) we robustly measure the intrinsic scatter of abundance ratios. At fixed [Fe/H] the RMS intrinsic scatter in [X/Fe] ranges from 0.04 dex (Cr) to 0.16 dex (Na), with a median of 0.08 dex. Scatter in [X/Mg] is similar, and accounting for [$α$/Fe] only reduces the overall scatter moderately. We consider several possible origins of the intrinsic scatter with particular attention to fluctuations in the relative enrichment by core-collapse supernovae (CCSN) and Type Ia supernovae (SNIa) and stochastic sampling of the CCSN progenitor mass distribution. The stochastic sampling scenario provides a good quantitative explanation of our data if the effective number of CCSN contributing to the enrichment of a typical sample star is $N \sim 50$. At the median metallicity of our sample, this interpretation implies that the CCSN ejecta are mixed over a gas mass $\sim 10^5 M_{\odot}$ before forming stars. The scatter of elemental abundance ratios is a powerful diagnostic test for simulations of star formation, feedback, and gas mixing in the early phases of the Galaxy.