We demonstrate a high-accuracy distributed fiber-optic temperature sensor using superconducting nanowire single-photon detectors and single-photon counting techniques. Our demonstration uses inexpensive single-mode fiber at standard telecommunications wavelengths as the sensing fiber, which enables extremely low-loss experiments and compatibility with existing fiber networks. We show that the uncertainty of the temperature measurement decreases with longer integration periods, but is ultimately limited by the calibration uncertainty. Temperature uncertainty on the order of 3 K is possible with spatial resolution of the order of 1 cm and integration period as small as 60 seconds. Also, we show that the measurement is subject to systematic uncertainties, such as polarization fading, which can be reduced with a polarization diversity receiver.