You do not need high acceleration, but more of a mild accelaration for a long time to reach high speeds. U can ride the solar wind or use ion rockets which are very efficient and can reach incredible speeds over a few months/years.
To escape from the Solar system you need to achieve at least like 50km/s which has been achieved by Voyager 1&2 !
On approaching spped of light, say 99%, the distances contract and YOUR time slows so you may quite well reach the destination alive within your lifetime. Of course u have to remember to start slowing down a few months early !
P.S. To understand the speed of light : It can circle the Earth 6 times in 1 second ! and 8 minutes from Sun to Earth...
U can use wormholes for even quicker and longer distances but those are a bit trickier. U need rotating black holes to survive and are not that predictable.....

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