Two Planks, Two Measurements

A carpenter must estimate the lengths $a$ and $b$ of two planks using a laser gauge whose each reading carries an independent, unbiased error of variance $\sigma^2$. The gauge's battery allows exactly two readings. The obvious plan — measure each plank once — leaves each estimate with variance $\sigma^2$. Can a cleverer pair of measurements deliver unbiased estimates of both lengths with lower variance, and how low can it go?