Given that, $ |x|+ |y| = 2 \quad \longrightarrow (1)$
Now, we can draw the figure.
The $ABCD $ is a rhombus with $AC$ and $BD$ diagonals whose length are $4.$
Area of rhombus $ ABCD = \frac{ AC\times BD}{2} = \frac{4 \times 4}{2} = 8\; \text{units}^{2}.$
Correct Answer $ : \text{C}$