jmc

Let the two random numbers be $x$ and $y$. In order to form an obtuse triangle, since 1 will be the longest side, we must simultaneously satisfy the following inequalities: $$x+y>1\text{ and }{x}^2+y^2<1.$$The first is the triangle inequality and the second guarantees that the triangle is obtuse. Graphing these in the $xy$-plane, we get the following shaded region. The curve is an arc of the unit circle centered at the origin. This area is then equal to that sector minus the right isosceles triangle within it, or $\frac{\pi }{4}-\frac{1}{2}=\frac{\pi -2}{4}.$ And since the area of the square is $1,$ $p=\frac{\pi -2}{4}.$

Four times $p$ is $\boxed{\pi -2}$.