We have measured the optical and mechanical loss of commercial silicon nitride membranes. We find that 50 nm thick, 1 mm² membranes have mechanical Q > 10⁶ at 293 K, and Q > 10⁷ at 300 mK, well above what has been observed in devices with comparable dimensions. The near-IR optical loss at 293 K is less than 2 x 10^-4. This combination of properties make these membranes attractive candidates for studying quantum effects in optomechanical systems.