1g constant acceleration will get you to relativistic speed (near C) in a little less than a year. Then there's the Delta V necessary to get there and the mass conversion/fuel efficiency issue. Short story is even if using Encapsulated Deuterium/Tritium fuel the energy conversion doesn't allow for any payload. It would all be fuel :-( *
Even if a fuel was available and all you had to deal with is a year accelerating/decelerating and add the rest of the distance/years there is another troubling issue. Space is not empty. Even interstellar gas would be an impediment akin to wind resistance. So drag increases as to the square of velocity ...
Anything larger than molecular would pass right through without a second thought depositing a fair amount of its mass as energy squared along the way.
https://en.wikipedia.org/wiki/Bussard_ramjet
Apologies to our favorite Fiction writers of the past century but the Bussard ramjet does not work with current technology or physics. It would however be able to vacuum ahead of the vehicle to remove any molecular debris and possibly "most of the interstellar dust up to several microns and process it through the vehicle as reaction mass and additional thrust. Think a kind of inverse deflector shield.
All in all it's an interesting topic but a bit of a dead issue for a technologically infantile society such as ours. Still it doesn't hurt to dream.
* fuel requirement is on the order of a metric shit ton. Nearly what you would find in the upper atmosphere of a gas giant.