9. For the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, the distance between the lines $A_{1} C_{1}$ and $B D_{1}$ is
Connect $B_{1} D_{1}$ and $A_{1} C_{1}$ intersecting at point $F$, connect $D F$ and $B D_{1}$ intersecting at point $E$. Given $\tan \angle B D_{1} D=\sqrt{2}$, $\tan \angle D_{1} D F=\frac{\sqrt{2}}{2} \Rightarrow B D_{1} \perp D F, A_{1} C_{1} \perp$ plane $B D D_{1} B_{1} \Rightarrow A_{1} C_{1} \perp B D_{1} \Rightarrow B D_{1} \perp$ plane $D A_{1} C_{1}$ $\Rightarrow E F$ is the common perpendicular segment of $A_{1} C_{1}$ and $B D_{1}$. In $\triangle E D_{1} F$, $\sin E D_{1} F=\frac{E F}{D_{1} F}=\frac{\sqrt{3}}{3}$ $\Rightarrow E F=\frac{\sqrt{3}}{3} \cdot \frac{\sqrt{2}}{2}=\frac{\sqrt{6}}{6}$. Therefore, the distance between the lines $A_{1} C_{1}$ and $B D_{1}$ is $\frac{\sqrt{6}}{6}$.