Accessing subwavelength information about a scene from the far-field without invasive near-field manipulations is a fundamental challenge in wave engineering. Yet it is well understood that the dwell time of waves in complex media sets the scale for the waves' sensitivity to perturbations. Modern coded-aperture imagers leverage the degrees of freedom (d.o.f.) offered by complex media as natural multiplexor but do not recognize and reap the fundamental difference between placing the object of interest outside or within the complex medium. Here, we show that the precision of localizing a subwavelength object can be improved by several orders of magnitude simply by enclosing it in its far field with a reverberant passive chaotic cavity. We identify deep learning as a suitable noise-robust tool to extract subwavelength localization information encoded in multiplexed measurements, achieving resolutions well beyond those available in the training data. We demonstrate our finding in the microwave domain: harnessing the configurational d.o.f. of a simple programmable metasurface, we localize a subwavelength object along a curved trajectory inside a chaotic cavity with a resolution of lambda/76 using intensity-only single-frequency single-pixel measurements. Our results may have important applications in photoacoustic imaging as well as humanmachine interaction based on reverberating elastic waves, sound, or microwaves.