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The human body can stand very fast speeds, but it can't cope with acceleration. To reach high speed you have to accelerate HARD. Otherwise it's weeks and months just to get moving quickly. And don't get me started on slowing down when you get there.....
And speed is relative to point of view. If you look at an airplane from the ground it moves pretty fast but for the people on board their reference is the plane itself and they're pretty much stationary. And, similarily to airplane, when craft speed or direction changes the force required affects to people on board too and that's where limits of the human body come up pretty quickly.
Acceleration at one gravity is 9.81ms^-1^ which is about 35km/h per second. That would take about a year to reach close to the speed of light.
You'd then need to spend the same amount of time slowing down.
So to get to our nearest stellar neighbour (alpha centaurii 4.3 LY distant) you still need to coast for about 2.3 years in the middle of the journey.
This is all assuming you can provide that much thrust constantly for two years.
We just need to discover astrophage.
See The Expanse if you want to see this in action at speeds significantly less than c.
Edit: Even for just getting around in the solar system, they had to magic up some chemicals to allow the human body to withstand deceleratory g-forces strong enough to keep travel times short enough to not completely wreck the narrative.