The pulleys and the string shown in the figure are smooth and of negligible mass. For the system to remain in equilibrium angle $\theta$ should be 

$45^\circ$

For mass m to be in equilibrium tension should be T = mg

vertical component of this tension should balance weight of mass $\sqrt 2 m$.

⇒ 2Tcos$\theta$ = $\sqrt 2 mg$

⇒ 2mg cos$\theta$ = $\sqrt 2 mg$

⇒ $cos\theta = \frac{1}{\sqrt 2}$ 

$\theta = 45^\circ$