Dyson shells are cute, just like Ringworld was cute.

Seriously, we will never be able to do anything like that.

We will be LUCKY if we manage to survive past the next global natural disaster that is coming. It could be any of several thing, including something nobody has ever thought of, but it certainly is headed our way.

Pick your favorite end of the world scenario here: http://www.exitmundi.nl/exitmundi.htm (some are silly, but fun anyway)

asteroid or comet impact

supervolcano

massive lava flows

new disease

major war (proof of human stupidity, as if any more were needed)

I want to point out that mass extinctions have been caused by such events before, and will doubtlessly happen again. It will be all we can do simply to survive the next mass extinction, let alone building stuff the size of a Dyson sphere. Instead of thinking about Dyson spheres, we need to build a system to detect and deflect incoming asteroids and comets. A supervolcanic eruption can not be prevented by any foreseeable extension of current technology, but who knows? We might be able to drill into one and release pressure slowly. Massive lava flows are something else entirely. The "Siberian Traps" covered much of Asia several thousand meters deep. The "Deccan Traps" did the same with much of India. I really can't see stopping a flow of two million cubic kilometers of molten lava, but it takes a long time to flow, and we might be able to think of something.

I would like to point out that the End Permian Extinction (likely caused by the Siberian eruptions) left the Earth so desolate, that there are as few fossils found in the first four million years of the Triassic rocks as in the Vendian rocks that preceded the Cambrian. Imagine a mass extinction that KEEPS live on the verge of extinction for four million years.

David F Mayer