jmc

The smallest right triangle with integer lengths is the $3-4-5$ right triangle. Since none of those lengths are multiples of 8, we must scale each side up by a factor of 8 to create a right triangle whose side lengths are all integer multiples of 8. This triangle has perimeter $3\cdot 8+4\cdot 8+5\cdot 8=(3+4+5)\cdot 8=12\cdot 8=\boxed{96}$ units.