Since it takes infinite energy to accelerate an object with mass to the speed of light, it can never happen. And you are right, if such an object did reach 99.99999...the speed of light, it would have mass approaching infinity relative to us.
Infinity in science means that making some parameter larger would not change the results. So if you have two H atoms, you know that they will be attracted to each other and readily form H2 molecule. However if they are beyond the range of the interactions, say 10-3 m, then it really does not matter if they are a cm apart or a light year, because they are separated by infinity. In this case infinity is 10-3 m. In other words, they will not form H2 when they are separated by infinity and in this case infinity is 10-3 m.
So infinity is relative to what you are discussing.
Let us suppose that a person is traveling at 99.999999 times the speed of light. Well if the forces acting on him or her are similar to those on us on Earth, they will feel just like us. This is why. It is relativity. In that person's frame of reference, he or she is not moving. Also if someone weighs 80 kg on Earth, he or she will also weight 80 kg in their frame moving 99.99999 time the speed of light relative to us. However from OUR frame, they have a mass which is near infinite.
But if he or she look at us, we are traveling at 99.999999 times the speed of light relative to them! Hence we look at them and, yes they are very heavy, moving very fast, and look very stretched in the direction of motion. But when they look at us see exactly the same..We move fast, we have infinite mass and we are stretched.
That was Albert's point.
Now another point. If you want to accelerate to 99.999999 times the speed of light (and have the energy to do it) and want to get to that speed within a normal life time, you would be crushed by the forces needed. On the other hand, if you accelerated at a rate that would not kill you, you would be long dead before getting anywhere close to the speed of light.