The emergence of movable antenna (MA) technology provides a promising way to enhance wireless sensing and communication by introducing spatial degrees of freedom through dynamic array reconfiguration. In near-field localization, achieving high resolution at low cost necessitates the adoption of sparse arrays. However, such sparsity tends to introduce spatial ambiguity due to aliasing effects. To resolve this resolution-ambiguity dilemma, this paper proposes an MA-enabled array zooming (AZ) system. First, we design a multi-measurement array zooming system that dynamically adjusts antenna spacings. By fusing the observational information from different measurements, the proposed AZ system effectively mitigates spatial aliasing while maintaining spatial resolution. Second, to quantify the performance limits under the severe multi-modal distributions inherent in sparse near-field sensing, we theoretically analyze the false peak distribution and derive a tighter performance lower bound, which incorporates the false detection probability. Third, considering that multiple false peaks may exist in practical multi-modal distributions, we propose an optimization algorithm for the AZ system to suppress false peaks and minimize the localization error. Extensive numerical results demonstrate that the proposed AZ strategy adaptively optimizes array configurations under varying signal-to-noise ratios (SNRs), substantially outperforming both conventional fixed-spacing arrays and Cramer-Rao bound (CRB)-based AZ benchmarks in localization accuracy.