Four reasons among many others it's not possible:
Weathervaning effect from the vertical surfaces of the tail would turn the aircraft as it accelerates, the tail would not remain upstream of the nose.
Taking off requires the engines to deliver their maximum power and the wing to convert this power into lift very efficiently to leave the ground. The poor efficiency of a reversed wing prevents generating required lift.
Thrust available from reversers is significantly weaker than regular thrust.
Takeoff rotation would be done with the nosewheel bearing all aircraft weight. Being not designed for this load, it would collapse.
Perhaps the most simple to understand difficulty is the aircraft tendency to weathervane, to behave like a weathercock when airspeed is significant.
As the vertical surfaces of the empennage are large, having them upstream of the engines makes the aircraft instable in yaw. As soon as the aerodynamic force is significant, the tail is slowed down, and this tendency, unless taken into account in the design, cannot be stopped before the tail is the rearmost flying part.
The more a multi-engine aircraft is powerful, the larger is the vertical empennage, in order to be able to maintain the desired yaw in case an engine is inoperative.
A wing is not reversible
Engines are designed to deliver their maximum thrust when an aircraft takes off (TOGA -takeoff / go around- setting). This maximal thrust must be efficiently used by the wing, which is required to have a given lift capability at this time. This is the case when the aircraft is accelerated in the regular direction.
(I would tend to say a wing cannot work at all in the reverse direction. It is fitted with asymmetric moving surfaces, likely to disintegrate when not loaded as intended, hinges being now at the wrong place. This is also valid for the tail surfaces. But let's assume everything remains in one piece.)
When airfoil edges are swapped, a condition known as reverse flow. Lift cannot be generated except at small angles of attack. This condition can happen on helicopter rotor blades and is classical for tidal turbines though the applicable Reynolds number is different.
The problem is the airflow does not attach well to the upper surface when the leading edge is sharp, stall occurs at small angles of attack. By itself the stall problem is a no go for taking off, as takeoff require a high angle of attack (15°).
If we want want to ignore this, we face a second problem: Reverse flow also introduces a dramatic loss of efficiency (L/D ratio). Lift is at least divided by two, and drag multiplied by two. This too is a condition preventing taking off. (Note the factor would probably be more like 10, but I've no reference to support this figure. More on that aspect. By the way this could make a good question and we have aerodynamics experts able to answer.)
To compensate lift loss takeoff speed needs to be increased, this requires more thrust and a longer runway to accelerate. Compensating for higher drag requires more thrust too.
At this stage, even with we had the full regular thrust, this wouldn't be enough to make the aircraft airborne. Instead of two engines we would need four or more.
Reverse thrust is limited
Engines provide reverse thrust at the expense of significant losses. Reversers design is very ineffective in term of aerodynamics as airflow is strongly deflected.
In addition only a fraction of thrust is reversed and some direct thrust is still generated. We can estimate the effective reverse thrust to be only 70% of TOGA.
While more than normal thrust would be needed, the thrust actually available is reduced and totally insufficient to take off.
When the engine is used in forward direction, air enters the engine pushed by ram pressure. When used in reverse direction, ram pressure is actually negative, air is constantly moved away from the inlet, the engine is not fed correctly, the compressor likely stalls and stops functioning before reaching the takeoff speed.
In a turbofan, the reverser reverses the fan flow, but the regular core flow is still required to spin the engine. It leaves the engine at the exhaust, which is now subject to ram pressure. The turbine is not designed to work in these conditions.
Aircraft attitude to take off
Now the visual part. In a regular takeoff, at the end of the acceleration roll aircraft nose is raised to increase the angle of attack (rotation phase), so the wing can generate the amount of lift required to leave the ground. When the aircraft is used in reverse gear, this gives this:
Aircraft image source, adapted
Assuming elevators are effective to lift the tail, which is not granted, there is an impossibility to balance the aircraft on the nosewheel during the roll and to keep it on the centerline, which is anyway not visible by the pilot.
The regular rotation is eased by the main gears location and the main gears are able to bear all the weight until the aircraft is airborne. In your dream the weight is now fully on the nose wheel, and the nosewheel is unable to resist, it collapses.
There are other points, which would be boring to explain but which are as significant, e.g. how to aerodynamically control the aircraft during the takeoff and in flight, as control surfaces wouldn't be movable under the reversed ram pressure.