This is what has been floating around in my head for a while now.
We're on a massive prothean ship that was discovered using the secret archives on Thessia, long before the events of ME1. The asari didn't reveal this to the rest of the galaxy until the reaper invasion. I guess they're that good at keeping secrets.
This ship is bigger than the collector ship (or maybe the base) featured in ME2. This ship, the 'Ark', has enough stasis pods to contain hundreds of thousand, if not millions of passengers. During the early events of ME3, the asari decide to announce this ship to the leaders of the species of the galaxy, with a plan. The plan is to pick the best and brightest of the galaxy (well, those who aren't working on other important projects, like the thing that goes into the Citadel [yes, I forgot what it's called]) and their families to be sent to another galaxy incase the Reapers succeed. However, this ship doesn't use a Mass Relay to travel so the journey to Andromeda takes many, many years.
Alternatively, it's basically the same as the above however there are no stasis pods and the trip takes many generations. Who we are in Andromeda are the distant descendants of those who originally left the Milky Way.
When this ship finally arrives, something happens to it which causes it to remain in the system it's in (which can explain why we're [apparently] not exploring the whole galaxy in this game). The ship is either stuck in a orbit around a planet, or landed on the planet. Either this big ship already had smaller ships attached to it, or the species of the Milky Way attached their own ships to it. Because this galaxy has no Mass Relay network, we're stuck with FTL or whatever to travel between words (again, assuming the game is isolated to one system, this would explain why) using these smaller craft.
The ship was filled with every species from the Milky Way. I'm unable to decide when the ship left. It needs to leave before the Genophage was cured (so there is no canon decision) but also have the geth and quarians on board. I don't know how to make that work. I haven't really put a whole lot of thought into this idea.
I have no idea about the state of the Andromeda galaxy, or what'll happen there. The above is just my little idea on how we get there.
Edit: I prefer the stasis option myself. There would be a VI/AI onboard similar to Vigil, who maintains everything.