jmc

Squaring both sides, we have $5|x|+8=x^2-16,$ or $5|x|=x^2-24.$ From here, we take cases on the sign of $x$:

If $x\ge 0,$ then we have $5x=x^2-24,$ so $$0=x^2-5x-24=(x-8)(x+3),$$which has roots $x=8$ and $x=-3.$ However, we assumed in this case that $x\ge 0,$ so we only get the solution $x=8.$

If $x<0,$ then we have $-5x=x^2-24,$ so $$0=x^2+5x-24=(x+8)(x-3),$$which has roots $x=-8$ and $x=3.$ However, we assumed in this case that $x\le 0,$ so we only get the solution $x=-8.$

We can check that both $x=8$ and $x=-8$ satisfy the original equation, so the product of the roots is $8\cdot -8=\boxed{-64}.$