Radars use time-of-flight measurement to infer the range to a distant target from its return's round-trip range delay. They typically transmit a high time-bandwidth product waveform and use pulse-compression reception to simultaneously achieve satisfactory range resolution and range accuracy under a peak transmitted-power constraint. Despite the many proposals for quantum radar, none have delineated the ultimate quantum limit on ranging accuracy. We derive that limit through continuous-time quantum analysis and show that quantum illumination ranging-a quantum pulse-compression radar that exploits the entanglement between a high time-bandwidth product transmitted signal pulse and and a high time-bandwidth product retained idler pulse-achieves that limit. We also show that quantum illumination ranging offers mean-squared range-delay accuracy that can be tens of dB better than that of a classical pulse-compression radar of the same pulse bandwidth and transmitted energy.