© 2019 American Physical Society. We theoretically investigate the tunneling of two-dimensional surface polaritons (2DSPs) through nanometer-wide gaps in atomically thin crystals. For quantitatively accurate results, we developed a rigorous model based on the diffraction of 2DSPs for strongly confined surface polaritons (i.e., the polariton wavelength much shorter than the free-pace photon wavelength). We find distinctive features of the tunneling of 2DSPs. First, radiation loss during the tunneling is shown to be negligible. Second, the reflection coefficient R and tunneling coefficient T are shown to exhibit an anomalous logarithm singularity in their dependency on the gap width. Even for a gap size over two orders of magnitude smaller than the surface polariton wavelength, an appreciable reflection coefficient was observed in our calculation. Finally, we show that when the gap size increases, the phase of R saturates very rapidly to a nontrivial value of π/4. Based on these results, we further examine resonant tunneling of 2DSP through two identical gaps separated by a distance L, and establish a resonance condition defined by L≈λsp(4n-1)/8 with a positive integer n.