Hydrogen-like light muonic ions, in which one negative muon replaces all of the electrons, are extremely sensitive probes of nuclear structure. Using pulsed laser spectroscopy, we have measured three 2S-2P transitions in the muonic helium-3 (μ³He⁺) ion, an ion formed by a negative muon and bare helium-3 nucleus. This allowed us to extract the Lamb shift, the 2P fine structure splitting, and the 2S-hyperfine splitting in μ³He⁺. Comparing these measurements with theory, we determined the root-mean-square charge radius of the helion (³He nucleus) to be r_h = 1.97007(94) fm, in good agreement with the value from elastic electron scattering but a factor 15 more accurate. Our results represent benchmarks for few-nucleon theories and open the way for precision quantum electrodynamics tests in He atoms and ions.