5. Given the hyperbola $C_{1}: 2 x^{2}-y^{2}=1$, and the ellipse $C_{2}$ : $4 x^{2}+y^{2}=1$. If $M$ and $N$ are moving points on the hyperbola $C_{1}$ and the ellipse $C_{2}$ respectively, $O$ is the origin, and $O M \perp O N$, then the distance from point $O$ to the line $M N$ is $\qquad$
5. $\frac{\sqrt{3}}{3}$.

Let the inclination angles of the lines $O M$ and $O N$ be $\alpha$ and $\beta$, respectively.
Substitute the points $M(|O M| \cos \alpha,|O M| \sin \alpha)$,
$$
N(|O N| \cos \beta,|O N| \sin \beta)
$$

into the equations of the hyperbola $C_{1}$ and the ellipse $C_{2}$, respectively, to get
$$
\begin{array}{l}
2|O M|^{2} \cos ^{2} \alpha-|O M|^{2} \sin ^{2} \alpha=1, \\
4|O N|^{2} \cos ^{2} \beta+|O N|^{2} \sin ^{2} \beta=1 .
\end{array}
$$

Then $\frac{1}{|O M|^{2}}+\frac{1}{|O N|^{2}}$
$$
=2 \cos ^{2} \alpha-\sin ^{2} \alpha+4 \cos ^{2} \beta+\sin ^{2} \beta \text {. }
$$

Since $O M \perp O N$, we have
$$
\begin{array}{l}
|\alpha-\beta|=\frac{\pi}{2}, \\
\cos ^{2} \beta=\sin ^{2} \alpha, \sin ^{2} \beta=\cos ^{2} \alpha . \\
\text { Therefore, } \frac{1}{|O M|^{2}}+\frac{1}{|O N|^{2}} \\
=2 \cos ^{2} \alpha-\sin ^{2} \alpha+4 \sin ^{2} \alpha+\cos ^{2} \alpha=3 \text {. }
\end{array}
$$

Let the distance from point $O$ to $M N$ be $h$.
$$
\begin{array}{l}
\text { Therefore, } \frac{1}{h^{2}}=\frac{1}{|O M|^{2}}+\frac{1}{|O N|^{2}}=3 \\
\Rightarrow h=\frac{\sqrt{3}}{3} .
\end{array}
$$