Answer
$S$ spans $R^2$.
Work Step by Step
Assume the combination
$$a(0,2)+b(1,4)=(0,0).$$
We have the system
\begin{align*}
b&=0\\
2a+4b&=0.
\end{align*}
The above system has only the trivial solution, that is, $$a=0, \quad b=0.$$
Consequently, $S$ is linearly independent and since $R^2$ has the dimension $2$ then $S$ spans $R^2$.