jmc

Any factor of $8000=2^6\cdot 5^3$ is in the form $2^a\cdot 5^b$ for $0\le a\le 6$ and $0\le b\le 3$. To count the number of perfect square factors, we must count the factors of $2^6\cdot 5^3$ that have $a=0$, $2$, $4$ or $6$ and $b=0$ or $2$. This gives $4\cdot 2=\boxed{8}$ perfect square factors.