It is impossible to actually go at the speed of light because in the formula for time dilatation (That's right Velocity distorts time. The faster you go, the slower time goes for you. This effect is negligible unless you are going very close to the speed of light.) if v(velocity) were equal to c(the speed of light) you would have a zero in the denomonator.
the formula is:
t'=t/sqrt(1-v^2/c^2)
where t' is the time that has passed from a stationary observer, t is the time that has passed for you, sqrt means square root and ^2 means squared. As you can notice, when v is small, since c^2 is so huge, you get a miniscule decimal, and 1 minus that number is basically as small as 1.0000000000001 or something, and so time is not really affected. As v gets larger(it can never be the speed of light, then the denomonator would be zero) that fraction becomes closer to 1, so when you subtract it from one, the denomonator gets smaller and smaller, meaning t gets smaller(in order for t' to remain constant) this means that time actual slows down for the object moving at such velocities. This formula is correct, it is part of Einstein's theory of relativity, and I have had to derive it myself from ordinary mechanics formulas.
Along with time, length contracts and mass expands by the same factor. If you were to go close to the speed of light, your mass would increase and length would decrease. This means that the density of your body would increase more and more as you went faster and faster, and your blood eventually wouldn't be able to flow anymore and your cells would die of oxygen starvation. Not that it really matters, because to even begin witnessing this effect, you would have to be going at least 90 or 95% or so the speed of light.
Now to answer your question in that is it actually physically possible to get close... considering they have particle accelerators stories high and miles long requiring tremendous amounts of energy in order to even get tiny protons and neutrons to move at such speeds, it doesn't seem like they're anywhere near accelerating something big anytime soon. F=ma, meaning in order to get something that weighs one gram, like a paperclip to accelerate to the same degree as they do the protons, you would need about 2x10^24 times the force. That notation just means a 2 with 24 zeros after it. The reason for this is that that is how much more mass a paperclip has then a proton. An actual human isn't going to get anywhere close for a very long time.