5. (C).
$$\begin{array}{l} \text { Let } x^{\prime}=x-59, y^{\prime}=y . \text { Then } \\ y_{1}^{\prime}=f\left(x^{\prime}+40\right), y_{2}^{\prime}=f\left(40-x^{\prime}\right) . \end{array}$$

Since the function $y=f(a+x)$ is always symmetric to the function $y=f(a-x)$ about the $y$-axis $(x=0)$,
then $y_{1}^{\prime} 、 y_{2}^{\prime}$ are symmetric about $x^{\prime}=0$.
That is, symmetric about the line $x=59$.