Solution:
Answer: $2^{2000}$
We claim that there is a bijection between pairs $(X, Y)$ and sets $S$ with at least $1001$ elements. To get $S$ from $X$ and $Y$, take $S = X \cup Y$, which contains $Y$ and thus has at least $1001$ elements. To form $(X, Y)$ from $S$, make $Y$ the largest $1001$ elements of $S$, and make $X$ everything except the largest $1000$ elements of $S$. Therefore we need to count the number of subsets of $\{1,2, \ldots, 2001\}$ with at least $1001$ elements. For every subset of $\{1,2, \ldots, 2001\}$, either it or its complement has at least $1001$ elements, so number of possible subsets is $\frac{1}{2} \cdot 2^{2001} = 2^{2000}$.