If you build a ship that accelerates at a constant 1g towards the destination until the midway point, then decelerates at a constant 1g, due to time dilation, the people on board would experience "only" 12 years passing for a 700LY journey. People on Earth would still need to wait 1400 years for the first communication back, of course, but at least you won't have to build generation ships.
The thing is, we aren't currently able to keep a stable acceleration of 1g.
The ship's mass will also increase relativistic-ally, as it will reach a maximum speed of ~0.9999961c, so more mass will require more force for the same acceleration.
...and according to that calculator, at 2gs it would take just over 7 years in the ship's frame of reference; at 3gs, slightly less than 5.
If you really want to get things over with quick, 10gs will get you there in well under 2 years. I wonder if that's survivable, if you were under water the whole time with a scuba tank or something?
The key to space is genetics. If we can master modifying our own code, we can solve all of the uncomfortable aspects of space travel (radiation, low/zero gravity, habitable environments※).
With extreme longevity you can even get over the time aspects (If a human can live for multiple centuries, a few decades spent on travel isn't that much time).
※ No need to terraform - instead of adapting the planet to humans, adapt humans to the planet. Scarce atmosphere? Add the Nepalese genes to survive at high altitude[1]. Heavy gravity? Increase bone and muscle mass. Water world? Gills it is!
