Does such a phenomenon exist? Can gravitational waves cancel each other out?
Gravitational waves when they are not strong compared to the background metric can interfere with each other, just like light can. They can form patterns of positive and negative (and everything in between) superposition, changing in space at distances on the order of the wavelength. Of course they won't cancel out throghout space (or maybe throughoput spacetime), just like electromagnetic waves also won't -- the energy can't dissapear. However, note that the gravitational field is a tensor field, not a vector field, and if one wanted 0 total field (in gravitation, in whatever coordinate system you were where you could do it all in the linearlized domain, all tensor components would have to cancel. For the example of the simplest one might visualize, for a circularly disposed set of particles, they alternately deform the circle in a sort of ellipse, with major and minor axis alternating. I'd think you could have another wave 180 degrees out of phase (or to reinforce it, in phase), but have not seen the math worked out.
Please note that this is in the weak field domain, meaning you can linearize the equations and consider the gravitational wave like an imagined spin 2 electromagnetic classical wave, on top of whatever your background metric was. If the field is stronger you'd need to take the nonlinear terms into account, and it gets more complicated.