Solution:
Let $X = \{x_1, x_2, \ldots, x_n\}$, $n \geq 1$, where $x_1 < x_2 < \cdots < x_n$.

If $x_n > 0$, then $x_n + |x_n| = 2x_n \in X$, which is a contradiction because $x_n < 2x_n$ but $x_n$ is the largest element of $X$.

The contradiction in the previous paragraph implies that $x_n \leq 0$. If $x_1 < 0$, then $x_1 + |x_1| = x_1 - x_1 = 0 \in X$. Hence, we must have $x_n = 0$, so that $x_i + |x_i| = x_i - x_i = 0 \in X$ for any $x_i \in X$, $i = 1, 2, \ldots, n$.

Hence, in order for the desired property to be satisfied, $X$ must be a finite subset of the interval $(-\infty, 0]$ and it must contain $0$. On the other hand, such subsets satisfy the said property.

The only nonempty finite sets that satisfy the desired property are those finite subsets of $(-\infty, 0]$ containing $0$.