Step 1: Angle condition:
Let the angles be $A \le B \le C$ with $C = 90^{\circ}$ and $B - A = C - B$. Then $A + C = 2B$, and with $A + B + C = 180$, $3B = 180$, so $B = 60$, $A = 30$.
Step 2: Use trigonometry:
The shortest side is opposite $30^{\circ}$ and equals 4. Then the hypotenuse is $4/\sin 30^{\circ} = 8$ and the other leg is $4/\tan 30^{\circ} = 4\sqrt{3}$.
Step 3: Area:
Half of the product of the two legs: $\frac12 \cdot 4 \cdot 4\sqrt3 = 8\sqrt3$.
Final Answer:
The area is 8 root 3 square units, option (B).
\[ \boxed{8\sqrt{3}\text{ sq. units}} \]