Chemical rockets are currently the only thing that allow sustaining such accelerations briefly for human-size payloads, but the low exhaust velocity and exponential reaction mass requirement make sustaining it for days/months/years completely impossible.
You'd have to supply the energy externally (i.e. some sort of beam propulsion), but getting any significant fraction of g out of such a system (with human-sized payloads) seems unlikely within the next centuries, especially as the distance increases.
Or some form of ram scope, plenty of H everywhere, but those also have the issue of you collecting things while going at relativistic speeds.
Magic warp bubble tech is the bare minimum, and we're nowhere close to inventing that.
Increasing the acceleration a little bit more than that (eg to 1%) would make a difference.
Wouldn't that be only a few months on a decades-long journey at 1.01G vs 1G??