I would say it is possible, but unlikely.
To build it outside the atmosphere of earth would be nigh impossible, as you would have to either source all of the materials from the planet and send them up into space, or get extraterrestrial materials and bring them back.
Either way it would have to be assembled in space, as you could not really part build the structure to assemble once it is in space in any reasonable capacity, and it would be too difficult to manufacture it once it is up there. So that option is out.
If it was within the atmosphere of earth, it would have to be held up by a structure, as outlined in the points here.
Therefore you would need every country and environmentalist to agree that it was in the best interests of the planet, as it would mean having giant structures sticking out of the ground within every country and ocean around the world.
Assuming geopolitical stability, second would be sourcing the materials. I would suggest carbon nanotubes, as they are very strong, can be very thin and there is plenty of carbon on Earth that can be refined. You again would need every country to contribute to sourcing the materials for this one.
Thirdly is the motion of Tectonic Plates. You cannot just have unmoving pillars going straight up all over the planet, as any movement in Uganda would affect the overall structure in Canada (Butterfly Effect). This would mean moving parts, so whilst the overall structure would be constant it could not be rigid.
You could have a hollow pipes-within-pipes kind of design. This would not only allow movement but also help assembly. You could not build a single constant shape, because it needs to be bigger than the Earth, so you wouldn't be able to. This then avoids the structural weaknesses you would generally find at the joints where separate pieces are attached together.
In terms of shape, I would suggest possibly tracking the lines of latitude/longitude. Kind of gives things a cagey look already.
As for weather, we have structures now that can withstand tornadoes/hurricanes, so it would be feasible. But considering it is right across the Earth, it would need to at least need to be maintained. Particularly if it needs to stand for a decade.
This again takes into consideration every country in the world allowing access to the structure from their territory.
Then again, there is always the second way that you could Cage Earth.