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At the moment there are deep seas and high mountains. But imagine that the land elevation of the Earth is equal everywhere. How deep would the ocean be in that case?

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An approximation can be obtained quite simply by dividing the volume of water in the oceans by the surface area of an ellipsoid with a smooth surface representing the idealized Earth in your question.

The volume of Earth's oceans, seas and bays is $1.332 \times 10^9 \text{ km}^3$.

The equatorial radius of Earth (semi-major axis of the spheroid) is $a = 6378.1 \text{ km}$. The polar radius of Earth (semi-minor axis) is $c = 6356.8 \text{ km}$.

The surface area of the oblate ($c < a$) spheroid is:

$$S = 2 \pi a^2 \left( 1 + \frac{1 - e^2}{e}\tanh^{-1} e \right)$$

where $e^2 = 1 - \frac{c^2}{a^2}$.

Which gives us $\approx 0.51 \times 10^9 \text{ km}^2$.

Dividing the volume of the oceans by this results gives us $\approx 2.6 \text{ km}$.

Note: Earth is not a sphere. An ellipsoid is a better representation of our Earth. Nevertheless, the answer to your question would have been approximately the same had I used a sphere instead, as suggested in the title of your question.

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    $\begingroup$ Do you think that because of all the mountains and seadephts the sizes you used (6378 and 6356) are wright? $\endgroup$ – Marijn Feb 3 '16 at 20:27
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    $\begingroup$ You can try and add the water of all lakes and lagoons, as well as the water in rivers and ice in glaciers and mountaintops, and see how much a variation you get. I am not much familiarized with the Snowball Earth hyphotesis. $\endgroup$ – carnendil Feb 3 '16 at 20:54
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    $\begingroup$ @userLTK -- It's the other way around. If this hypothesis is correct, ice sheets up to 3 km thick ice covered the continents, but not the oceans. The oceans would have been covered by a much thinner layer of ice, possibly with partially open water near the equator. $\endgroup$ – David Hammen Feb 7 '16 at 14:15
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    $\begingroup$ @DavidHammen Thanks, and that makes sense. Ice would build up on land but perhaps stay liquid in the salty oceans, with, as you said, a thinner ice layer on top. Sea level must have dropped perhaps as much as a mile in that case. $\endgroup$ – userLTK Feb 8 '16 at 4:40
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    $\begingroup$ Even with a smooth Earth, the oceans wouldn't be a uniform depth. For the same reason Earth is an oblate spheroid, the Earth + water would be a slightly larger oblate spheroid. The ocean would be shallower at the poles and deeper at the equator by the same ratio as the Earth's axes, so 1/290. $\endgroup$ – kwknowles Apr 11 '16 at 3:43
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510,100,000 square kilometers of surface area, and a total of 1,386,000,000 cubic kilometers of water gives you a 2.717 kilometer column of water across the whole planet if it was billiard ball smooth, but the same basic shape.

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    $\begingroup$ I should note that there will be some variance, in the range of metres due to tidal and rotational effects, and very very minor variances due to local thermal conditions. $\endgroup$ – Ash Aug 16 '17 at 11:16
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    $\begingroup$ Hopefully, you mean ideally snooth rather than the much rougher surface of a billiard ball. Earth is already smoother than a billiard ball. $\endgroup$ – Spencer Feb 11 '18 at 1:16

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