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:
Square fold http://cs.smith.edu/%7Eorourke/MathOverflow/OctFold.jpg
For more realistic physical models, the shape would be more bag-like, and the analysis more difficult.
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:
Square fold http://cs.smith.edu/%7Eorourke/MathOverflow/OctFold.jpg
For more realistic physical models, the shape would be more bag-like, and the analysis more difficult.
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:
For more realistic physical models, the shape would be more bag-like, and the analysis more difficult.