Two sticks, two measurements

Idea. Both measurements involve both sticks, so each length is estimated from two measurements instead of one. Averaging two independent errors halves the variance.

Step 1: the estimate. With $S=a+b+e_1$ and $D=a-b+e_2$, $$\hat{a}=\frac{S+D}{2}=a+\frac{e_1+e_2}{2}.$$

Step 2: its variance. $$\mathrm{Var}(\hat{a})=\frac{{\sigma }^2+{\sigma }^2}{4}=\frac{{\sigma }^2}{2}.$$ So the answer is $\frac{1}{2}$, and the same holds for $\hat{b}=\frac{S-D}{2}$.

Compare. Measuring $a$ directly gives variance ${\sigma }^2$. The combined design gives each length the precision of two measurements, with the same total effort. This idea, measuring combinations rather than single items, is the basis of experimental design and of weighing problems.