Solution:

A radius-$1$ disk covers segment $\overline{AB}$ exactly when its center is within $1$ of both $A$ and $B.$ That region $R$ is the lens where the two unit circles centered at $A$ and $B$ overlap.

Each unit circle passes through the other's center, so the lens is bounded by two $120^\circ$ arcs. Two $120^\circ$ sectors of area $\tfrac{\pi}{3}$ overlap in two equilateral triangles of total area $\tfrac{\sqrt3}{2},$ giving $R$ area $\tfrac{2\pi}{3} - \tfrac{\sqrt3}{2}.$

The set $S$ consists of all points within $1$ of $R.$ Beyond $R$ itself, this adds two $60^\circ$ sectors of radius $1$ (each area $\tfrac{\pi}{6}$) and two $120^\circ$ annuli of outer radius $2$ and inner radius $1$ (each area $\pi$).

Therefore the area of $S$ is $$\left(\tfrac{2\pi}{3} - \tfrac{\sqrt3}{2}\right) + 2 \cdot \tfrac{\pi}{6} + 2\pi = 3\pi - \tfrac{\sqrt3}{2}.$$

Thus, the correct answer is C.