This is confusing. I have often heard that energy is conserved and it neither be created nor be destroyed. But, if a body is accelerated in space, neglecting every celestial body, it would never stop accelerating and would gain kinetic energy forever as speed is increasing too. And this would disprove the law of conservation of energy. But I suspect that somewhere I am wrong or maybe everywhere. Please clarify my doubt.
No, objects will not accelerate forever. To have acceleration, you must have a force (from $F = ma$) and just being in space doesn't create a force.
What does stay constant in space (more technical word: vacuum) is speed, since there's no friction, no gravity, etc to slow down the body. If speed remains constant then kinetic energy remains constant as well.
If a body is accelerating, it must mean it is being driven by a force F or it converts it's mass into energy.
If it is being driven by a force, the work done by the force translates into the kinetic energy, and if it uses its own mass for acceleration then at some point it's mass becomes zero ( the mass translates into kinetic energy).
Energy is conserved in both the cases. A body will accelerate only till it is given energy which it converts into kinetic energy.
Let us say we have a force field which is constantly acting on the imaginary body. In this case the body will accelerate until it reaches the speed of light. Special theory of relativity states that no mass can travel faster than the speed of light. So it will not and can not accelerate infinitely.
A body will only accelerate when there is some force acting on it. If the force applies for a definite time (eg a boost from the rocket) the acceleration stops when the force stops. In any case, whatever extra energy that is gained by the body being accelerated is lost by the source of the force- the gains and losses cancel out so that energy is conserved.