If you were able to construct a spacecraft that accelerated constantly at about #1#g for #14# years (ship time), then decelerated constantly at about #1#g for #14# years (ship time), then that would be sufficient to not only leave our galaxy, but get to the neighbouring major galaxy, the Andromeda Galaxy, approximately #2.5# million light years away.
One downside is that by the time you could return (another #28# years of ship time), a total of about #5# million years would have passed on Earth. So not only would all your ground support team have died, but humans might have evolved into something else or at least become extinct in the meantime.
How about the more modest goal of just leaving the galaxy?
We are about #25000# light years from the centre of our galaxy, which has radius about #50000# to #60000# light years. In other words, we are about halfway out, with the nearest point on "the rim" (so to speak) about #25000# to #30000# light years away. So, regardless of how hard we try, it would take at least #25000# years of ground team time for us to reach the edge of the Milky Way, since it is (probably) not possible to travel faster than light.
Like the Andromeda trip, ship time could be significantly shorter, but it's still not much use.
What about just travelling out of the plane of the galaxy? It's relatively flat isn't it? Well yes, that would cut our journey down to a mere #500# or so light years, but that would still take at least #500# years of ground team time.
The other downside of travelling at high speed is the difficulty of shielding the ship against the effects of hitting small particles on the way, which would effectively irradiate the contents of the ship, including the travellers.