https://gregsspacecalculations.blogspot.com/p/blog-page.html
1G of acceleration (which I'd hope you agree is survivable by humans) over an extended time period can easily reach relativistic speeds.
1 day .0028c
1 week .02c
1 month .086c
1 year .77c
2 years .97c
3 years .996c
4 years .9995c
5 years .9999c
The thing stopping us from doing this today is economics, not physics. Current rockets have about enough fuel for minutes of acceleration, and fuel requirements increase exponentially due to the tyranny of the rocket equation. If you skip the need for fuel (laser propulsion?) and find some way to decelerate (laser cooling propulsion???), then interstellar travel to pretty much anywhere becomes entirely reasonable within human lifespans.Honestly, Star Trek way back in the 1960s was pretty brilliant at getting around many of these technical problems by inventing "warp drive".
Even if we somehow magically solved our economic problems overnight, that isn't going to make relativistic speeds feasible for humans anytime soon, if ever.
We really can skip the need for fuel, for example. Sails (with or without lasers) are a technology we have proven in the field. Lasers do lose collimation over distance, but you can reach relativistic speeds before then (I'd argue that .02c from 1wk of 1G is relativistic). That won't get you to the center of the galaxy (or solve the deceleration issue), but there are proposals being reviewed today to use lasers to send probes for a flyby of Proxima Centauri.
https://www.nasa.gov/general/swarming-proxima-centauri/
https://en.wikipedia.org/wiki/Breakthrough_Starshot
But we don't need science fiction tech for this to work, we just need impractical amounts of fuel. Starship only has enough fuel to last 10m? Just send 50k starships and you can burn for a year. Tyranny of the rocket equation requires additional fuel to push all that fuel? Just send another billion Starships or whatever. Going too fast and now the interstellar medium hits like high-energy cosmic rays? Just send more shielding and fuel. This assumes we can build and fuel billions of Starships, which is certainly infeasible, but I'm calling this an economics issue as we have these technologies today.
If we want to get really sci-fi, I'd point you towards stellar engines. The thought process here is that the Earth already provides radiation shielding, and the Sun already burns fuel to provide massive amounts of energy, so we might as well just make use of what we got! Add mirrors to concentrate the Sun's light in one direction, and our entire solar system becomes an interstellar spaceship. It might take millions or even billions of years, but the Sun has enough fuel to accelerate the whole system to about .27c.
Nothing with mass can have the same speed as light, but you can trivially accelerate a human body - or something similar - to a speed which is arbitrarily close to it, without risking anything from the G-forces involved.
You just need to do it very slowly.
That is, in any case, neither here nor there, since this is a thought experiment used in a discussion about the effects of moving at a speed close to c - people in thought experiments are stronk.