The radius of the smallest circle with centre at the origin and having non-empty intersection with $S=\left[\left(x, y\right):\left|x\right|+2\left|y\right|=1\right]$ is radius of inscribed circle of rhombus having sides represented by equation $\left|x\right|+2\left|y\right|=1$.

$r=\frac{|0+2\left(0\right)-1|}{\sqrt{1+4}}=\frac{1}{\sqrt{5}}$