No, there is no way that could get accomplished, unless we had a series of absurdly huge scienctific breakthroughs. I would hazard a guess that we could not get 1000 people to mars with 60 years of intense world effort.
We could get 1,000 people to Mars by the end of 2030 if we really wanted to, if money was no object and every effort was put to it. Possibly sooner.
The issue would be maximising their long-term survival, though, once they get there, and ensuring they even land safely. But since money would be no object we would undoubtedly over-engineer everything and make it as safe as possible.
But yeah, do-able for sure. And the issue isn't the number of people but really just how quick you can build the things. If you can launch one ship with 10 people then you can build another 99 in relatively quick order, I'm sure, especially with every effort and no concern for financial viability.
A billion people, though? Heck no. That's over 45,000 a day each and every day for 60 years lifting off from Earth.
But what size would it need to be? Well, i think the closest thing we have would be cruise ships. But bear in mind that these have limited resources on board, and could survive for only a short time, not the permanent duration of any Battlestar.
Anyhoo, the largest cruise ship I believe is the MS Harmony of the Seas... Can carry c.8500-9000 people (including crew), and is roughly 360 by 60 by 20m. I've never been on a cruise ship, but let's assume that that is as crowded as you'd want it to be.
Scaling it up for 1 billion people you would need a ship roughly 48 km^3 - so 4km by 4km by 3km, so roughly the double the size of the largest Borg cubes.
The weight of the cruise ship c.120,000 tonnes, so for 1 billion people you'd need c.13,000 MILLION tonnes of mass in orbit.
Saturn 5 launched c.140,000 kg into LEO, so that's 140 tonnes.
It would therefore take c.100 million launches to get that mass into orbit.
That's c.4,500 launches per day for 60 years.
But as said, a cruise ship has enough provisions for maybe 10 days (?) so on top of all this you would need equipment and livestock to be able to be self sufficient. Would you need to double, triple the size? I don't know.
Let's assume that we develop some fast growing seaweed or some such that can resolve much of our hunger, we can at least use the corpses of the 10 million or so who die each year to help provide the crop with nutrients, but it's not going to be enough. There's all the recycling equipment as well, even if we should perfect a long-lasting system.
Then there's the issue of production of the ships, not even considering needing to use more launches to get people up there, building it, and being able to stay up there for a length of time.
Now, maybe you could build large chunks on earth and lift them into orbit, but I'm not sure what the maximum payload theoretically is for our current technology.
Anyhow, raw materials: if the ship is to be made of, say, steel then it would consume, say, 10 billion tonnes. Per wiki the world currently produces c.1,600 million tonnes a year, so over 60 years this would be 96 billion tonnes. Not enough for your 12 ships, and that's just using the cruise-ship as the model.
So no, not possible, I don't think.
