Since $|a| = 5$, it follows that $a = \pm 5$.

Since $a + b = 0$, it follows that $b = -a$, which means $b = \pm 5$.

There are two cases to consider:
1. When $a = 5$, $b = -5$, then $a - b = 10$;
2. When $a = -5$, $b = 5$, then $a - b = -10$;

Therefore, the answer is $\boxed{\pm 10}$.