As shown in the figure, by symmetry, when the section is perpendicular to the diagonal $B D_{1}$ (or $A C_{1}$), the section is a regular hexagon. At this time, the area is $S_{1}=6 \cdot \frac{\sqrt{3}}{4} \cdot\left(\frac{\sqrt{2}}{2}\right)^{2}=\frac{3 \sqrt{3}}{4}$; also note that the area of the diagonal section of the cube is $S_{2}=\sqrt{2}$, and $\sqrt{2}>\frac{3 \sqrt{3}}{4} \Rightarrow 4 \sqrt{2}>3 \sqrt{3} \Rightarrow 32>27$, which is clearly true. Therefore, the maximum section area is $\sqrt{2}$.
5. The maximum cross-sectional area of a cube with edge length 1, when cut by a plane, is $\qquad$