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.
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.