Basically, our frail human bodies are limited to maybe only 2G's (I'm spitballing, I would appreciate any real studies) of sustained acceleration, tops. That means Andromeda is a biiiiig nope.
Going to a web-page that someone (in this thread?) mentioned, rather than working it all out myself, I plugged in the biggest value it would apparently allow for distance ("999999999999999" light-years, essentially 1x10
15the Andromeda Galaxy being 'merely' 2.5x10
6) and a 2g acceleration (that being for half the journey, an identical deceleration for the rest), a human would have to experience a 'mere' 34 years and two months (plus change) on that ship, in their largely relativistic travel to the destination. (A few orders of magnitude of distance, either way, appear not to make the waiting time vary significantly in this circumstance.) So a 30yo would get there before they hit retirement age. Assuming no personal or ship-wide problems en-route.
Plug in a more 'friendly' 1g acceleration/deceleration rate and Andromeda's (approximate) distance, and we're talking about 'cruising' for 28 years, seven months, 1.5-ish weeks. Give or take the approximations (and, Andromeda is heading towards us, as well, isn't it? ...but not sure if significantly so for such a journey) I think that's not a problem for a (single-)Generation Ship. The kids and grand-kids (or more, if the residents are bored and have nothing
else to do) could get back home as well, albeit to a world changed by time and/or swallowed up by the Sun. That is
assuming that one can maintain the 1g thrust for that period, of course, for the size of ship that can contain the corresponding thrusters and fuel-sources of sufficient longevity.
(And I'm taking those figures on trust, but it seemed to work for far nearer, less relativistic calculations that I'd previously laboriously made off my own back, when I was first shown it.)