Long-lived alpha and beta emitters in the 222Rn decay chain on detector surfaces may be the limiting background in many experiments attempting to detect dark matter or neutrinoless double beta decay. Removal of tens of microns of material via electropolishing has been shown to be effective at removing radon daughters implanted into material surfaces. Some applications, however, require the removal of uniform and significantly smaller thicknesses. Here, we demonstrate that electropolishing < 1 μm from stainless-steel plates reduces the contamination efficiently, by a factor > 100. Examination of electropolished wires with a scanning electron microscope confirms that the thickness removed is reproducible and reasonably uniform. Together, these tests demonstrate the effectiveness of removal of radon daughters for a proposed low-radiation, multi-wire proportional chamber (the BetaCage), without compromising the screener's energy resolution. More generally, electropolishing thin layers of stainless steel may effectively remove radon daughters without compromising precision-machined parts.