5. In a regular quadrilateral pyramid $P-ABCD$, it is known that the side faces are equilateral triangles with a side length of 1, and $M, N$ are the midpoints of edges $AB, BC$ respectively. Then the distance between the skew lines $MN$ and $PC$ is $\qquad$.
5. $\frac{\sqrt{2}}{4}$.

As shown in Figure 2, let the diagonals $A C$ and $B D$ of the base intersect at point $O$. Draw a perpendicular from point $C$ to line $M N$, intersecting $M N$ at point $H$. Since $P O$ is the perpendicular to the base, $P O \perp C H$.
Also, $A C \perp C H$, thus $C H$ is perpendicular to the plane $P O C$.
Therefore, $C H \perp P C$.
Hence, $C H$ is the common perpendicular segment between line $M N$ and $P C$.
Note that, $C H=\frac{\sqrt{2}}{2} C N=\frac{\sqrt{2}}{4}$.
Thus, the distance between the skew lines $M N$ and $P C$ is $\frac{\sqrt{2}}{4}$.