We report the validation of multiple planets transiting the nearby (d=12.8pc) K5V dwarf HD 101581 (GJ 435, TOI-6276, TIC 397362481). This system consists of at least two Earth-size planets whose orbits are near a mutual 4:3 mean-motion resonance, HD 101581 b (R_p_=0.956_-0.061_^+0.063^R_{Earth}, P=4.47d) and HD 101581c (R_p=0.990_-0.070_^+0.070^R_{Earth}, P=6.21d). Both planets were discovered in Sectors 63 and 64 TESS observations and statistically validated with supporting ground-based follow-up. We also identify a signal that probably originates from a third transiting planet, TOI-6276.03 (R_p=0.982_-0.098_^+0.114^R_{Earth}, P=7.87d). These planets are remarkably uniform in size and their orbits are evenly spaced, representing a prime example of the "peas-in-a-pod" architecture seen in other compact multiplanet systems. At V=7.77, HD 101581 is the brightest star known to host multiple transiting planets smaller than 1.5R{Earth}_. HD 101581 is a promising system for atmospheric characterization and comparative planetology of small planets.