Solution:
12.5

Explanation:
I am especially fond of these two ways to solve this problem:

1. Draw the right triangle whose hypotenuse is the square's diagonal, and whose two legs are two sides of the square.  Then use the Pythagorean Theorem (a^2 + b^2 = c^2) to solve for the length of each side.  Since two sides are equal, we get (a^2 + a^2 = c^2), or (2(a^2) = c^2) ).  Since c is 5, 2(a^2) = 25, making a^2 equal to 25/2, or 12.5.  Since the area of the square is a^2, we're done: it's 12.5.

2. Tilt the square 45 degrees and draw a square around it such the the corners of the original square just touch the middles of the sides of the new, larger square.  The new square has sides each 5 units long (the diagonal of the smaller square), and it therefore has area 25.  However, a closer inspection reveals that the area of the larger square must be exactly twice that of the smaller.  Therefore the smaller square has area 25/2, or 12.5.