All right, I guess if I must. I know nothing about this topic, and all I ever manage to do is confuse everyone. But I enjoy it, so...

Whether you feel that it is possible to travel faster than the speed of light, or not this is simply physically IMPOSSIBLE. Once again, the most conservative estimates of human existence is about 6000 years, but, lets just to simplify that to 5000 as it’s a nicer number. If we have been accelerating for 5000 years at 9.8m/s/s we would currently have the velocity of 1.5379x10^11 which is much faster than the speed of light. If faster than light travel were possible, we know that it would lead to travel backward through time.

You are absolutely correct that according to special relativity we cannot move faster than the speed of light. The reason for this being that as your speed increases, your relativistic mass increases as well. If you want to keep accelerating at the same rate, you have to keep adding energy. The faster you go, the more energy you have to add. If you do not keep adding energy, your rate of acceleration will decrease. Your mass will keep increasing (and thus your acceleration decreasing) at such a rate that to pass c you would need an infinite amount of energy. (Engineer, stop me if any of this is wrong. This is just my very basic understanding)

Now of course, at this point you're likely to saying to yourself "Aha! The acceleration is decreasing, therefore the acceleration of "gravity" must be decreasing as well! This isn't happening, therefore the theory is false!"

The reason you are saying this of course, is because you didn't pay attention. Had you paid attention you would have noticed I said "relativistic mass". Your mass increase is relative. To someone to whom you are not moving near c relative to, your mass is equal to your rest mass, therefore the energy required to accelerate you hasn't increased.

To someone here on Earth, Earth's speed never increases, therefore Earth's relativistic mass never increases, and therefore the energy required to accelerate Earth never increases. Assuming the "universal accelerator" never runs out, relative to us Earth will accelerate at a constant rate indefinitely without ever approaching c, much less passing it.

P.S Note my liberal use of the word "therefore". I think this word looks important, and I like that; therefore I will be peppering and salting it throughout my posts in the future.