According to calculations, most of the way to the very edge of the universe could be traveled in a human lifetime.
All you need is a rocket capable of accelerating at 9 metres per second per second. You would be traveling at speeds close to the speed of light, so time would be slower for you due to relativity.
Because the universe is expanding, and because that expansion is accelerating, the expansion horizon could never be reached, but you could get 99 percent of the way in 50 years.
Or, as it turns out, in even less time:
[Juliana Kwan at the University of Sydney in New South Wales, Australia] and her colleagues have found the trip could take even less time.
Based on the latest cosmological values for dark energy and other parameters, they showed an astronaut could make the journey in only 30 years.
But their calculations also suggest that returning home presents its own challenges. Even slight uncertainties in the strength of dark energy or the total density of matter in the universe could cause a spacecraft to miss Earth by millions of light years.
Beginning the deceleration just a second too late could cause you to overshoot the Milky Way, Kwan says. “You would effectively be lost in space.” [New Scientist]
Not that it matters. Even if you made it back, 70 billion years would have passed on Earth and nothing you remember would remain- even the Sun would be gone.