First let us examine the orientation of the three rooms. We start in room $A$ with our back towards the entrance. As we step through the right door into room $B$ our orientation changes by an angle $R$. From there going through the right door into room $C$ means another rotation over an angle $R$. Finally we go through the left door, over an angle $L$, into room $A$. Note that our orientation is the same as the begin situation, since our back is once again oriented to the entrance of room $A$. So we have: $R + R + L = n*360$ degrees, with $n$ an integer. Now it is given that $L$ is opposite $R$, hence $L = R + 180$. It follows that $R$ is either $60$ degrees or $180$ degrees. The last possibility leads to a contradiction: as we exit room $B$ we return to room $A$, however it is given that we enter room $C$. Therefore we can exclude an angle of $180$ degrees. Conclusion: room $B$ must be rotated $60$ degrees with respect to $A$, and similarly room $C$ is rotated $60$ degrees with respect to $B$.
The next step is to find the location of the doors. Let us denote the position of the entrance of room $A$ by $(0,0)$ and the right door by $(1,0)$. We then find that the entrance of room $B$ is $(1,0)$ and the right door is $(3/2, 1/2 \sqrt3)$. For room $C$ we find that the entrance is $(3/2, 1/2 \sqrt3)$, the left door is $(0,0)$, and the right door is $(1, \sqrt3)$. Finally we obtain that the left door in room $A$ is located at $(0, \sqrt3)$ and the left door in room $B$ is at $(-1/2, 1/2 \sqrt3)$. To summarize: the six doors are located at the corners of a regular hexagon, which has its centre at $(1/2, 1/2 \sqrt(3)$.
We should now be able to allocate the space in and around the hexagon to the three rooms. We can indeed connect the entrance of each room to the right door. However this is not possible when it comes to the left doors! The three left doors are positioned such that they interfere with each other. Hence no solution (in $2$ dimensions) is possible.