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The problem is that a continuous one g acceleration will still not allow you to exceed the speed of light (c).

Wild speculation warning:
If we could ever develop something analogous to Star Trek's "beam me up" technology, so we could convert people to an EM signal and transmit them somewhere at lightspeed, then the transit time for the travelers would seem instantaneous. All we'd need is a reception station at the other end (something Star Trek somehow seems to avoid).

To create a network of reception stations, we'd need to send ships the slow way, perhaps they'd be robotic, and have them assemble receivers at desired (or even random) locations. That's the first phase, and it would take a few million years to get this infrastructure built over a good chunk of the galaxy.

But when it's finally built -- actually while it's being built, as far as it extends -- we could be traveling around as EM signals. Such transmissions might not look all that different from natural light, so they'd be undetectable, unless we were looking for them.

"If we could ever develop something analogous to Star Trek's "beam me up" technology, so we could convert people to an EM signal and transmit them somewhere at lightspeed, then the transit time for the travelers would seem instantaneous. All we'd need is a reception station at the other end (something Star Trek somehow seems to avoid)."

Aargh.

There are two ways to do this: (1) "record" the position and state of every atom in the transportee and send the information. At the other end (how did you get there?) you have a big tank of atoms and build them up into an exact copy of the guy back home. Now you have two guys, both with the same SSN and bank account, same DNA, same fingerprints...so you murder the original. Hmm.

(2) You convert the victim into energy and send him that way. That's gonna hurt. Also there is a distinct danger of blowing up the Earth because a 100-kg person bears (E=mc2) 9x10¹⁸ joules of energy. A big catcher's mitt to catch that beam...

But even worse, once you catch it, you have to DEAL with it; convert it (somehow) back into mass. Presumably the instructions for making the person ride the BEAM on a subcarrier or something. Catchers' mitt better be big and good, otherwise you will vaporize the target planet.

And both schemes leave you (the transportee) at the mercy of the inverse-square law...you're melting, melting (like the Wicked Witch of the West) with distance, pretty soon the "BEAM" looks like a flashlight...then a firefly...then...poof! eaten by the redshift. Better take out some insurance either way!

--Boris