Nope, it's far more simple than that, thanks to Einstein's discoveries. All we need is a ship that can produce a 1g thrust over a long time. This is still a big technological jump from where we are now, but it's well within the realms of known physics. With a 1g thrust we could be on the other side of the galaxy, 100,000 light years away, in just over 22 years of ship time. That's relatively easy, no pun intended, and doesn't require any fancy new physics. The Earth time for the journey will be around 100,071 years, so there's no coming back or sending lots of emails home, but if we simply want to get humans out there it's not a problem.
The bonus is a 1g acceleration solves a lot of the medical problems like muscle (inc heart) wastage and mineral leaching.