First let me clarify: You propose four moving surfaces which extend as needed (I guess that is what you mean by "open from the fuselage") from the fuselage?
Yes, theoretically it can work, but the flaps need to be as big as the surfaces they are intended to replace and need to be out in the airstream all the time. That would look remarkably like what we currently have, so any hopes of significantly reducing drag are unfounded.
Now there is a scenario where drag savings on the flaps are possible. Tail surfaces don't work at their maximum or even optimum lift (optimum meaning at their best lift to drag ratio) most of the time, so the degree of extension could be smaller, and less surface area is exposed. But where do you stow all the remaining part of the flap? Now you need to make the fuselage that much bigger, so you lose in fuselage drag what you gain in tail area drag. Add to this the weight of the kinematics for moving the flaps, and the conventional design looks ever more attractive.
As @fooot has pointed out, this scenario needs computer control and added layers of safety like secondary means of flap movement if the primary actuators fail.
If no asymmetric thrust scenario is possible, earlier designs have already proposed to do away with the whole vertical tail and replace it with a small, computer-controlled fin at the nose, yielding significant weight and drag savings.
Aircraft designers have constantly tried to reduce tail surfaces to the minimum they can get away with. Please look at the picture of the An-70 below closely: The horizontal tail has slats which open forward and up, so the tail surface can produce more downforce when flaps are extended. This is a trick to reduce the tail area, which doesn't need to produce much downforce during most of the flight, at the expense of more drag with flaps extended and more moving parts.
An-70 at Paris Airshow in 2000 (own work)