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Rotationally supported, cold, gaseous disks are ubiquitous in astrophysics and appear in a diverse set of systems, such as protoplanetary disks, accretion disks around black holes, or large spiral galaxies. Capturing the gas dynamics accurately in these systems is challenging in numerical simulations due to the low sound speed compared to the bulk velocity of the gas, the resolution limitations of full disk models, and the fact that numerical noise can easily source spurious growth of fluid instabilities if not suppressed sufficiently well, negatively interfering with real physical instabilities present in such disks (like the magneto-rotational instability). Here we implement the so-called shearing-box approximation in the moving-mesh code AREPO in order to facilitate achieving high resolution in local regions of differentially rotating disks and to address these problems. While our new approach offers manifest translational invariance across the shearing-box boundaries and offers continuous local adaptivity, we demonstrate that the unstructured mesh of AREPO introduces unwanted levels of "grid-noise" in the default version of the code. We show that this can be rectified by high-order integrations of the flux over mesh boundaries. With our new techniques we obtain highly accurate results for shearing-box calculations of the magneto-rotational instability that are superior to other Lagrangian techniques. These improvements are also of value for other applications of the code that feature strong shear flows.