Why haven't we yet tried accelerating a space station with people inside to a near light speed?
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 $5.67 \cdot 10^{20}\text{ J}$.)
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!
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!