There are fairly good theoretical engineering designs for space-craft, mainly City-farms with sustainable populations of inhabitants - "generation ships" - that could travel at speeds ranging up to one tenth of the speed of light.
At that incredible speed, this galaxy is "
only" a mere three hundred and fifty million years away.
Had we started just before the dinosaurs walked the planet, we could be arriving about now.
Of course, there are also valid reasons for thinking that friction with the interstellar gas and dust would restrict speeds in real space to less than one hundredth of the speed of light no matter how good our engineering is.
If that is so, then this galaxy is three
thousand five hundred million years away and to arrive now we would have had to start our journey not very long after the Sun was born when the Earth was still coallescing. Were we to start "now" [meaning sometime within the next million years or so, small delays like that wouldn't make much of a difference in this endeavour] we would arrive just before Sol becomes a Red Giant and pasteurises this planet.
Three and a half milliards of years is a long bus-ride even for a generation ship. It may not be possible to keep a habitat habitable for so long.
This cosmos is rather large and absent magical faster than light machinery almost all of it will forever be out of reach of us and our descendants no matter how our technologies develop.