Solution:

Each interior angle is $120^\circ,$ so extending sides $FA$ and $BC,$ $BC$ and $DE,$ and $DE$ and $FA$ cuts off three equilateral corner triangles and forms a large equilateral triangle.

The corner triangles built on $AB, CD,$ and $EF$ are equilateral, and one finds the large triangle has side $1 + 4 + 2 = 7,$ while the removed triangles have sides $1, 2,$ and $1.$ The area is $$\frac{\sqrt3}{4}\left(7^2 - 1^2 - 2^2 - 1^2\right) = \frac{43\sqrt3}{4}.$$

Thus, the correct answer is E.