6. Inside a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, there is a small ball that is tangent to the diagonal segment $A C_{1}$. Then the maximum radius of the ball is $\qquad$
6. $\frac{4-\sqrt{6}}{5}$.

When the radius is maximized, the small sphere is tangent to three faces of the cube. Without loss of generality, assume the small sphere is tangent to the three faces passing through point $D_{1}$. Establish a spatial rectangular coordinate system with $D_{1}$ as the origin, $\overrightarrow{D_{1} C_{1}}$, $\overrightarrow{D_{1} A_{1}}$, and $\overrightarrow{D_{1} D}$ as the positive directions of the $x$, $y$, and $z$ axes, respectively.
Let $A(0,1,1)$, $C_{1}(1,0,0)$,
and the center of the small sphere $P(r, r, r)$.
Then the distance from point $P$ to $A C_{1}$ is
$$
\frac{\left|\overrightarrow{A P} \times \overrightarrow{A C_{1}}\right|}{\left|\overrightarrow{A C_{1}}\right|}=\sqrt{\frac{2}{3}}|1-2 r|=r \text {. }
$$

From $r<\frac{1}{2}$, we get $r=\frac{4-\sqrt{6}}{5}$.