We show how the finite sizes of unordered defect cores in discretized orientation and vector fields can reliably be estimated using a robustness measure for topological defects. Topological defects, or singular points, in vector and orientation fields are considered in applications from material science to life sciences to fingerprint recognition. Their identification from discretized two-dimensional fields must deal with discontinuities, since the estimated topological charge jumps in (half-)integer steps upon orientation changes above a certain threshold. We use a recently proposed robustness measure [Hoffmann & Sbalzarini, Phys. Rev. E 103(1), 012602 (2021)] that exploits this effect to quantify the influence of noise in a vector field, and of the path chosen for defect estimation, on the detection reliability in two-dimensional discrete domains. Here, we show how this robustness measure can be used to quantify the sizes of unordered regions surrounding a defect, which are known as unordered cores. We suggest that the size of an unordered core can be identified as the smallest path radius of sufficient robustness. The resulting robust core-size estimation complements singular point and index estimation and may serve as uncertainty quantification of defect localization, or as an additional feature for defect characterization.