jmc

Since the square has side length $2/\pi$, the diameter of each circular section is $2/\pi$. The boundary of the region consists of 4 semicircles, whose total perimeter is twice the circumference of a circle having diameter $2/\pi$. Hence the perimeter of the region is $$2\cdot \left(\pi \cdot \frac{2}{\pi }\right)=\boxed{4}.$$