Here we report atomic structure calculations and experimental observations of the two-photon 6s² ¹S₀ → 6s23s ¹S₀ transition in Yb. The unique level structure of Yb places the 6s6p ¹P₁ state's energy at ≈49.7% of the ionization energy, which facilitates two-photon Doppler-free excitation of the 6s23s ¹S₀ Rydberg state from the 6s² ¹S₀ ground state at a relatively low optical intensity. This J= 0 → J=0 transition is immune to first-order Zeeman and electric quadrupole shifts, has reduced sensitivity to higher-order Zeeman shifts, and is further predicted to have relatively low sensitivity to density shifts. Using configuration interaction theory, we compute relevant properties of the 6s23s ¹S₀ Rydberg state for sensing, including the C₆ coefficient, dc polarizability, blackbody radiation (BBR) shift, and BBR depopulation rate. As a first experimental step, we demonstrate two-photon Doppler-free spectroscopy of the 6s² ¹S₀ → 6s23s ¹S₀ transition. From our experimental spectra, we estimate the natural linewidth (lifetime) of the 6s23s ¹S₀ state to be 90 kHz (1.8 μs). We believe this transition, or other 6s² ¹S₀ → 6sns ¹S₀ transitions in Yb with n≈23, may find use in a variety of sensing applications, potentially including a two-photon clock, electrometer, or BBR thermometer.