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This sort of thing is done, already, (on a very small scale) by chemical engineers, who bolt together atoms and make complex molecules. Any fool can make good copies of CO2 molecules - just by breathing out!
I think we would need quantum computers for the vast quantities of data required - I suspect with modern hard drive technology the disk would need to be light years across!
There's also something in quantum mechanics called the "no cloning theorem," thatbasically says that you can't copy a quantum state exactly. But like others have said, if all you want to do is make something with similar properties as the thing you're trying to copy, that isn't a problem. If you want an exact replicate, that would be impossible.