Interesting fact: When people say 'the sky,' they usually mean light, color, clouds, and atmosphere all at once, even though those are not the same thing physically.
The answer: The sky itself is not a single object with one fixed weight, but if you mean the atmosphere that creates the visible sky, it weighs about 5.15 x 10^18 kilograms globally. If you mean only the blue light you see, that light has energy but no rest mass.
This is a great example of how everyday language and physics diverge. The sky is hard to weigh because part of it is material atmosphere and part of it is simply light scattering through that air. The question matters because it pushes us to separate appearance from substance, which is exactly how science turns vague impressions into measurable quantities.
