Let the squirrel be at height \( x \). The rod bends by angle \( \theta \). The horizontal displacement of the center of mass is approximately \( x\sin\theta \approx x\theta \).

The gravitational torque attempting to topple the rod is: \[ \tau_{gravity} = mg(x\theta) \]

The elastic restoring torque opposing the bend is: \[ \tau_{restoring} = C\theta \]

For the rod to remain stable (and not buckle continuously), the restoring torque must exceed the gravitational torque:

\[ \tau_{restoring} \ge \tau_{gravity} \]

\[ C\theta \ge mg x \theta \]

\[ x \le \frac{C}{mg} \]

Substitute the given value \( C = 2mgL \):

\[ x \le \frac{2mgL}{mg} = 2L \]

This means the rod is mechanically stable for a squirrel up to height \( 2L \). Since the actual physical length of the rod is \( L \), the squirrel can safely climb the entire length.