jmc

The vertex of the parabola is the origin. Let $A=\left(a,\frac{a^2}{8}\right)$ be one vertex of the triangle. Then by symmetry, $B=\left(-a,\frac{a^2}{8}\right)$ is another vertex of the triangle.



Then $A{B}^2=(a+a{)}^2=4a^2,$ and $$O{A}^2=a^2+\frac{a^4}{64}.$$Hence, $4a^2=a^2+\frac{a^4}{64}.$ Then $3a^2=\frac{a^4}{64},$ so $a^2=3\cdot 64,$ which means $a=8\sqrt{3}.$

Therefore, the side length of the triangle is $2a=\boxed{16\sqrt{3}}.$