Can $S^4$ be the cotangent bundle of a manifold? I am asking the question because in the classical mechanics book by Arnold, he states that there is a distinguished $1$-form on $T^*V$. It seems that there is no such distinguished $1$-form on a general even-dimensional manifold. So, not every even-dimensional manifold can be the phase space of a system, or not every even-dimensional manifold can be a cotangent bundle? Is $S^2$ a cotangent bundle (intuitively it is not)? How about $S^4$?

As I alluded to in the comments, vector bundles of positive rank are necessarily non-compact. Therefore, no compact manifold can be the total space of a vector bundle of positive rank over any topological space. In particular, neither $S^2$ nor $S^4$ can be a cotangent bundle.

On the other hand, any space can be viewed as a rank zero vector bundle over itself.