1G of thrust is impossible to maintain for long without millions of times more energy than humans have ever generated during the entirety of human history. The most dense amount of energy possible is antimatter, which we've only created nano-grams of at a cost that would extrapolate to $67 trillion per gram.

And if the math on this page is correct,

https://www.desy.de/user/projects/Physics/Relativity/SR/rocket.html

for every ton of ship & payload you want to send to Proxima Centauri you need 10 tons of antimatter to get there and 37 tons to actually slow down to visit instead of flying right by. The reason for the difference is the tyranny of the rocket equation, which means every ton of fuel you add to try to go faster increases the amount of fuel you need to get to the same speed because you also have to accelerate that additional fuel And that assumes 100% perfect engine efficiency, which is unlikely.

In reality solar sails driven by huge lasers is going to be an important part of travel to nearby stars because it avoids the tyranny of the rocket equation.