How wide would a structure have to be so that it would be stable enough to reach space?
If you are thinking along the lines of a space elevator, the structure would not need to be huge, maybe a couple of hundred metres across, maybe a few tens of metres. In simple terms, the space elevator concept means that a mass at the upper end is in orbit and holds up the main shaft. The shaft would therefore be mainly loaded in tension. The bigger the shaft, the greater the resistance to wind passage therefore the higher the lateral loads, so it would be advantageous to minimise the shaft diameter. The biggest problem with commercially available material technology is that there isn’t really anything strong enough. Suitable materials are only in the lab, are very expensive and are not made in sufficient quantities. Potential materials include carbon nanotubes and synthetic spider silk.