jmc

Since $D$ is the midpoint of $\overline{AB},$ it has coordinates $$\left(\frac{1}{2}(0+0),\frac{1}{2}(0+6)\right)=(0,3).$$The line passing through $C$ and $D$ has slope $$\frac{3-0}{0-8}=-\frac{3}{8};$$the $y$-intercept of this line is the $y$-coordinate of point $D,$ or $3.$ Therefore, the equation of the line passing through points $C$ and $D$ is $$y=-\frac{3}{8}x+3;$$the sum of the slope and $y$-intercept is then $$-\frac{3}{8}+3=-\frac{3}{8}+\frac{24}{8}=\boxed{\frac{21}{8}}.$$