Every smooth manifold $M$ can be realized as a Lagrangian submanifold of a symplectic manifold (for example, inside its own cotangent bundle $T^*M$ carrying the standard symplectic form $\omega_0$).
From this question/answer it looks as though every smooth (compact) manifold can be realized as a Lagrangian submanifold of a Kahler manifold by making a neighborhood of the zero section of the cotangent bundle Kahler.
What about realizing a smooth manifold as a Lagrangian submanifold of a Calabi-Yau manifold? I would guess this is more restrictive (for example, one probably can't always put a Calabi-Yau structure on a neighborhood in the cotangnet bundle), but is much known about this?