I would say a resounding: Maybe
Here are the challenges that can probably be overcome with the motivation and funds.
The first issue is the time frame.
From the movie: When Worlds Collide
You can waste anything but time!
It takes about 10 years just to design and prototype a new space mission. Then launching starts. This can be shortened if frequent high loss tests are allowed.
There would also have to be massively parallel and massively replicated design and build programs.
You need parallel development because you don't have time to start over.
You need to replicate each build in multiple locations because you will need massive launches of the programs that succeed. Also, you don't want to have a situation where the one program that might succeed got knocked out do to an accident.
For example, have 10 different project designs of which are each different project is being built in 10 different locations (essentially, 100 different, simultaneous Apollo programs)
By throwing money and people at it, you can probably get it built in 10 years
The second issue is the biosphere
We know from Biosphere 2 that we don't really know what we are doing when building biospheres.
Can we figure it out by then? Maybe.
At the same time that the project is being designed and lifters are being built, we can run biosphere tests here on Earth. Most will likely fail. Those that fail early enough can be restarted. However, at best, we will learn which techniques make a biosphere stable over 10 years.
Again, massive parallel and massive redundant will help.
Another biosphere issue is that it will need to be huge (3.14 acres for 8 people). Just look at the amount of space, materials (structure, dirt, water, air, plants & animals) that went into it. Lifting all that will be a tremendous project even once you know what to lift.
The third issue is that it will be too close
An asteroid that size will throw up a lot of debris. The arc will have to get out of LEO. That will take fuel. If you have it boost out to the Moon for a bit and then come back to an Earth orbit, you need to ship a lot of fuel on board. I would boost to a Lunar orbit, circularize around the Moon, and stay until most of the debris has fallen back to Earth. Then boost back to Earth and re-circularize the orbit.
I would have it come back to Earth for these reasons:
I wouldn't trust the number of required thrusters needed to move such a pig 1000 years after they were built. The people 1000 years in the future might not be able to make any real repairs if they forget enough.
I would want the space that the fuel was in for expanding the biosphere.
Can you have a biosphere without gravity? Biospheres generally have a water cycle. It's hard to have flowing rivers without gravity. So we need to spin it. This isn't a deal breaker but it is one more thing that will add cost, materials, and ways to go wrong.
- Who gets to go.
- Who has to build it.
- What will the people who don't get to go do?
Just look at the politics that go into our current space program. The parts Shuttle was built in 3 different timezones because politicians threatened to block approval unless they got specific jobs in there districts. Then the parts were shipped for assembly. Can you imagine if there were lives at stake?
Also under politics, think of terrorist threats.
It could work if everything goes right but I don't think that it is likely.