The answer is: nothing
What If Everyone Jumped at Once?
Eight billion people is an enormous number of people and a negligible amount of planet.
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Step 1 of 5
Pushing off the ground does push the ground.
This part is real. When you jump, you push Earth down exactly as hard as it pushes you up. The reason nothing happens is not that the push is missing — it is that Earth is heavy.
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Step 2 of 5
Compare the two weights honestly.
Drawn to scale, the second bar would be invisible. All the humans alive weigh roughly 400 million tonnes. Earth weighs about 6,000,000,000,000,000,000,000 tonnes. Humanity is about one ten-trillionth of the planet we are standing on.
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Step 3 of 5
So the planet moves less than the width of an atom.
Work it through and Earth shifts by roughly a hundredth of the width of a single hydrogen atom. There is no instrument that could detect it, because it is smaller than the things instruments are made of.
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Step 4 of 5
And then it moves straight back.
Everyone comes down again. Landing pushes Earth back up by exactly the same amount that jumping pushed it down. The net effect at the end is precisely zero — not approximately zero.
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Step 5 of 5
The dangerous part is the gathering, not the jumping.
Squeezing eight billion people into one place is the genuinely catastrophic step. Shoulder to shoulder, humanity fits in a city the size of Los Angeles — and then has to get home with no food and no roads.
This is a good habit for any big-number claim: check the ratio before you check the drama.
The short version
Everyone jumping moves Earth by a fraction of an atom's width, and landing puts it back — humanity is a ten-trillionth of the planet.
Try it yourself
Jump on a bathroom scale and watch the needle spike. That spike is the push. Earth gets exactly that, shared out over six sextillion tonnes.