While it takes light approximately 2.54 million years to traverse the gulf of space between Earth and, for instance, the Andromeda Galaxy, it would take a much shorter amount of time from the point of view of a traveler at close to the speed of light due to the effects of time dilation; the time experienced by the traveler depending both on velocity (anything less than the speed of light) and distance traveled (length contraction). Intergalactic travel for humans is therefore possible, in theory, from the point of view of the traveller.
Accelerating to speeds closer to the speed of light with a relativistic rocket would allow the on-ship travel time to be drastically lower, but would require very large amounts of energy. A way to do this is space travel using constant acceleration. Traveling to the Andromeda Galaxy, 2 million light years away, would take 28 years on-ship time with a constant acceleration of 1g and a deceleration of 1g after reaching half way, to be able to stop.
Unfortunately the amount of fuel required is so large it would make this impossible with today’s technology.