By Borel and Serre 1950 (Impossibilité de fibrer un espace euclidien par des fibres compactes. (French))
It is impossible to have this situation of fiber bundles: $\mathbb R^n\rightarrow B$ with compact fibers $F$, unless $F$ is a point.
So now suppose that $\mathbb R^n$ is foliated by compact leaves $\{L_\alpha\}_{\alpha\in \Delta}$ then the quotient space $B:=\mathbb R^n/\{L_\alpha\}_{\alpha\in \Delta}$ is Hausdorff and we have the fiber bundle $\mathbb R^n\rightarrow B$ with compact fibers! is this a contradiction of the assumption that $\mathbb R^n$ is foliated by compact leaves?