Solution:

Suppose that it can be done. Color the building in checkerboard fashion, and suppose that we obtain $a$ white rooms and $b$ black rooms. Each door connects a white room with a black room. So, if we consider, for each white room, the number of doors adjoining it, we will count each door exactly once. Since every room is to have 2 doors, the total number of doors will be $2a$ by this count. But similarly, if we consider, for each black room, the number of adjoining doors, each door will be counted once, so that the total number of doors is also equal to $2b$. So, $2a = 2b$, or $a = b$. It follows that the total number of rooms is $a + b = 2a$, an even number. But we also know the number of rooms is $2001^2$, an odd number - contradiction. Hence, the desired condition cannot be met.