This is a very interesting question. We always have this kind of question when we learn that there is a light barrier for any real matter. To be honest, I am not an expert on relativity, I still do not have a very good answer to it. But I can share some of my thoughts.

Before I start, I would like to make an assumption, the light barrier or speed of light should be stated in vacuum. Since it is known, the speed of light in medium is slower than that in vacuum. Suppose we have some type of special medium, such that the light barrier in this medium is very small, then this is not the question you want to know.

1) Light barrier: Einstein proposed one of the principles for special relativity: no real matter can break the light barrier in vacuum. It seems to be true so far, since a lot of experiments have been done, and verified Einstein's special and relativity theories. Thus for most of us we believe that the principle is most likely to be true.

2) Things travel at the speed of light or even faster. Of course, light itself is traveling at the exact speed, which is the standard we are comparing to. Even though, no real matter can travel faster than light, there are things can travel faster than light. Some example, phase velocity and quantum entangled information. It is probably not easy to explain these terms in just a few words. The bottom line, these things traveling faster than light are not real matter, because there are information that can spread faster than light. Photon (light) can travel at the speed of light, because its rest mass is zero.

(There are other possibilities that an object can travel faster than light, for instance the assumption of warping the space. But I am not going to explore the details.)

3) Back to the original question:
First, can a person break the light barrier? Very unlikely. The special relativity imposes the following relation
E = m c² /(1 - v²/c²)^0.5,
where E is the energy, m is the rest mass of an object, c is the speed of light in vacuum and v is the velocity that the object travels.

With v = 0, we have the famous relation
E = m c².

If you accelerate v from 0 to c, the energy will increase, and then eventually to infinity.

Second, what if a person breaks the light barrier, what happens? Unless you have no mass, meaning no existence of you; or you survive at such high speed somehow, carry an infinite amount of energy, equaling or even greater than the total energy of the universe. Now you won't even notice the existence of time. Everything looks eternal to you.

The last, everything I mentioned before is based on what we learnt at small speed, from 0 to the light barrier speed c. It is possible that there is a different structure, or different universe that everything moves larger than c, which breaks the light barrier. So how would it look like? The picture could be totally different than we described before. This is something very interesting to me either. But I don't think we will get a correct answer to it in the near future. Keep your curiosity and hope we can get a good answer in the future.