When you put the pieces of the field together, you have a rectangle that's 53 yards 
wide and 120 yards long.
When you draw the diagonal, it divides the field into two right triangles.Each one 
has legs with lengths of 120 yards and 53 yards. The diagonal of the field is the 
hypotenuse of both right triangles. 
(120)² + (53)² = (the hypotenuse)²
14,400 + 2,809 = (the hypotenuse)² 
17,209 = (the hypotenuse)²
The diagonal (hypotenuse) = √17,209 = 131.183... yards (rounded)
I guess that's the longest straight line it is possible to run on a football field 
while staying in bounds.