Helium atoms in pulsed supersonic beams have been prepared in the circular Rydberg state with principal quantum number
n=55 using the crossed-fields method. High-resolution microwave spectroscopy of the transition from this state to the
n=56 circular state, at frequencies close to 38.5 GHz, was performed to measure static magnetic and electric fields along the axis of propagation of the beams with quantum-state-selective detection by pulsed-electric-field ionization. Magnetic fields of between 1.3 and 1.6 mT were measured to a relative precision of $\ifmmode\pm\else\textpm\fi{}900\phantom{\rule{0.28em}{0ex}}\mathrm{nT}$ by rf spectroscopy and $\ifmmode\pm\else\textpm\fi{}1.3\phantom{\rule{0.28em}{0ex}}\textmu{}\mathrm{T}$ by microwave spectroscopy, with absolute calibration, accounting for Doppler shifts and effects of weak stray electric fields to $\ifmmode\pm\else\textpm\fi{}2.0\phantom{\rule{4pt}{0ex}}\textmu{}\mathrm{T}$ and a spatial resolution of $\ifmmode\pm\else\textpm\fi{}0.87$ mm. Magnetic-field gradients could be determined to a precision of $\ifmmode\pm\else\textpm\fi{}1.49\phantom{\rule{0.28em}{0ex}}\textmu{}\mathrm{T}/\mathrm{mm}$ ($\ifmmode\pm\else\textpm\fi{}53$ nT/mm) over a baseline of 1.74 mm (35 mm). To perform these measurements, static electric fields and contributions from the motional Stark effect were minimized, and residual electric fields in each of the three spatial dimensions in the apparatus were measured to an absolute precision of between $\ifmmode\pm\else\textpm\fi{}85$ and $\ifmmode\pm\else\textpm\fi{}750\phantom{\rule{0.28em}{0ex}}\textmu{}\mathrm{V}/\mathrm{cm}$. The methods used in this work can be transferred to experiments with other atoms or molecules. They are therefore well suited for applications in minimally invasive, absolute static-field magnetometry and electrometry, for example, at hybrid interfaces between Rydberg atoms and superconducting circuits; in tests of bound-state QED or the weak equivalence principle with atomic or molecular hydrogen, antihydrogen, or positronium; and in measurements of the absolute neutrino mass by cyclotron radiation emission spectroscopy.