If I understand the question properly, the answer depends on the flexibility of the sheet.
A fine cloth napkin might crease almost like origami, in which case the height would be half the diagonal of the square,
$\sqrt{2}/2$ for a unit square:
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![Square fold][1]
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For more realistic physical models, the shape would be more bag-like, and the analysis more difficult.


  [1]: http://cs.smith.edu/~orourke/MathOverflow/OctFold.jpg