Why haven't we yet tried accelerating a space station with people inside to a near light speed? Is that something we could do if we use ion or nuclear thrusters?
Wouldn't people in the station reach 0.99993 speed of light in just 5 years accelerating at 1g and effectively travel into the future by 83.7 years? 
That would be a great experiment and a very effective way to show relativity theory in action. I mean, the people inside the station would have effectively traveled into the future, how cool is that? Why haven't it been done yet?
 A: It is not feasible because it would cost an enormous amount of energy
to accelerate the spacecraft.
To prove this let's calculate with some concrete numbers.
Very optimistically estimated, your spacecraft may have a mass of $m=1000\text{ kg}$ (enough for a few people and a small space capsule around them, but neglecting the mass of the fuel needed).
And you said you want a speed of $v=0.99993\cdot c$.
Now you can calculate the relativistic kinetic energy of it:
$$\begin{align}
 E_{\text k} &= \frac{mc^2}{\sqrt{1-v^2/c^2}} - mc^2 \\
   &= \left(\frac{1}{\sqrt{1-v^2/c^2}}-1\right) mc^2 \\
   &= \left(\frac{1}{\sqrt{1-0.99993^2}}-1\right)\cdot 1000 \text{ kg}\cdot (3\cdot 10^8\text{ m/s})^2 \\
   &= (84.5-1)\cdot 1000 \text{ kg}\cdot (3\cdot 10^8\text{ m/s})^2 \\
   &= 7.5 \cdot 10^{21}\text{ J}
\end{align}$$
Now this is an enormous amount of energy.
It is comparable to the yearly total world energy supply.
(According to Wikipedia:World energy consumption
the total primary energy supply for the year 2013 was $5.67 \cdot 10^{20}\text{ J}$.)
A: In addition to the other conceptual aspects, there are practical ones.
If you accelerate at 1g for 5 years, then you'll end up x light years away, where x is going to be tedious to compute (I assume you integrate tanh((9.81 m/s^2)*(5 years)/c)), but is clearly going to be measured in *light-years*.
So you're clearly enitrely isolated from the earth, and in the middle of space.
You can't actually do anything interesting with the fact that you're "in the future", because out in space nothing's really happened. In order for it to be at all interesting you'd have to come back to Earth.
But you're also travelling at 0.999934479 c. Away from Earth
Now you have to turn around, and decelerate for 5 years to stop, and then spend another 10 years coming back.
So this 5 year trip, has turned into a 20 year trip, during which your space station has to be entirely self-sufficient ... and not have any problems.
We just don't have the systems to support an environment like that for that long.
Then you've got the psychological and sociological issues to take into account.
Plus the fact that you can't steer this thing.

I can imagine all of these problems being soluble, if we can create a high-10s of personnel space station.
But a) we haven't currently achieved that and I don't think we're anywhere plausibly near achieving that, and b) suddenly the numbers in Thomas Fritsch's answer aren't starting from 1,000kg, they're starting 1,000 tonnes or more!
A: I'm no physicist, but, just to add to the list of insurmountable problems with this idea, I've always thought the hardest problem was the "air resistance" in space.
The density of interstellar space is about 1 atom per cubic centimeter. If your spaceship is 1 meter cubed, and travels at c for 1 second, you have travelled 300,000 kilometers, encountering 300 trillion atoms.
When you are moving at relativistic speeds, each proton you run into is delivering 0.003 joules of energy into you. For the above distance, that's 900 GJ. 100 seconds in, and you have experienced pushback equivalent to a nuclear bomb.
Things are a little bit better in the intergalactic medium, where the density is 1 atom per cubic meter, a million times less than in regular interstellar space. That means 900 MJ per second of travel. That's 1 ton of TNT every 5 seconds. Whew, much better!
I'm not even taking into account the possibility that fusion will be undergone for many of these atoms on the surface of your spaceship. Good luck finding a material that can withstand that.
I'm super amateur so I may be miscalculating here, please correct me if I am!
A: Mission time.
One of the issues of a manned Mars missions is the time window for an optimal transfer. I takes about six months to get to Mars. If you want to leave for the Earth again using the same window, you have about 10 days to visit Mars.
That's the equivalent of flying from London to New York, visiting the tourist shop for 30 minutes, and then flying back.
But the next window is about 1.5 years later. If you take this window, including travel time to and from, that the mission will take about 2.5 years in isolation and close quarters.
We have no accurate data on the effect of being isolated and contained for such a period, and it is not as easy as you think. There is no way of reaching them or providing supplies. They need to be self-contained and handle all obstacles (e.g. mechanical failures) themselves.  
Your proposed experiment would take twice as long. What you've also not accounted for is that the craft needs to slow down again, which would take another 5 years, making the whole mission four times longer than a Mars mission, which is already stretching the boundaries of what we are able to reliably achieve.
It's even worse than a Mars mission, because a Mars mission at least has radio contact. But as the crew gets exposed to time dilation (which will be noticeable much before they reach light speed) communication will become less feasible, and they will be truly alone.

Distance.
Travelling at high velocity (it doesn't even need to be near lightspeed) means that this craft cannot orbit the Earth anymore. So where will it travel? Well, you'd expect them to be about 2.5 lightyears removed from us (on average, they travelled at half of lightspeed during their 5 year speed up = 2.5 lightyears).
But don't forget to account for the slowdown. That's another 2.5 lightyears before they come to a "standstill" (if there were such a thing).
To put it into perspective, that's about 120% the distance to Alpha Centauri.

What's the point?
Okay, so let's say we accomplish all that and send the mission. Then what? They are in the future, out past Alpha Centauri, with no hopes of coming back and dwindling supplies. We can't find out what they experienced. We can't find out anything. There is no benefit to us whatsoever. 
This is pretty much the equivalent of shooting a single person in a spacesuit in a particular direction with no comms or any way to get back. You're just sending someone out to a lonely death. You're not accomplishing anything.
And given the amount of time they will have traversed (keep in mind that time dilation will happen relative to their speed, so they will already experience more than half the time dilation for the last 2.5 years of their speed up time), it's likely that the humans alive then will have found other and better way to confirm relativistic effects.  
So either the astronauts sacrificed themselves without contributing to the civilization that sent them; or they get picked up by a civilization that has already figured out the thing they gave their lives in pursuit of finding out. Neither is a good outcome.
A: The only current  propulsion  systems, that I know of, that could achieve high speeds is the same one they've been talking about for decades, setting off nuclear bombs behind a pusher plate. However, I heard somewhere that this would only be feasible for about 10% of the speed of light. And, like in the other answers, you would have to have protection from incoming atoms in space.
