I'm not sure why everyone seems to think it would automatically be a huge project on the scale of the crucible to travel to a different galaxy. There are lots of good non-monetary reasons why you would have technology that could get you there, but it never gets used.
This is currently the case for interstellar travel in real life. We have technology that could get big groups of people between star systems, with estimated costs low enough that a megacorp could potentially build one. We don't do this because:
1) It would cost $400 billion+ dollars, which is pricey but still under half a year of US defense budget spending. It would completely drain a moderately sized group of billionaires to build privately, and it is unlikely you'll find such a group that would be willing to throw away their entire fortunes on the endeavor in our current environment
2) The most viable methods for doing this involve building large stockpiles of essentially nuclear weapons to use as fuel for the trip, so the most likely current outcome would be getting arrested by the NSA for attempting to privately build nukes
3) You would have a decent chance of blowing up and killing everyone on board, because at $400 billion/shot, you can't really afford to make full scale test rockets, and initial rocket designs often fail despite all the computer modeling
4) It would take generations to get there, so the people that launch will all die on that ship in space, and their decendants would be the ones to possibly colonize a new system
5) If you don't end up having a habitable planet when you get there, everyone on the ship will eventually die because there won't be fuel to go anywhere else
6) You'll kill a few people on earth with radioactive fallout from launching the ship, and may fry electronics from EMP in a localized area
These are all really good reasons not to build an orion drive and go flying off to try and expand humanity to the stars, which is why we haven't done it. But we do know how to do it and it could be done with a relatively small fraction of earth's GDP if there was a sufficiently strong motivator. You can bet that multiple individuals or groups would build an interstellar ark if our solar system came under an existential threat. It is entirely reasonable that the civilizations in the mass effect universe know how to get to Andromeda but have never tried because of similar reasons. In particular, spending a sum of money that could purchase a large corporation on an endeavor where everyone currently buying into it will be dead before it does anything is a tough sell in attracting investors.