Milnor-Stasheff "Characteristic classes" problem 2-C says:
Any vector bundle over a paracompact base space can be given a Euclidean metric
in other words, if $\pi : E \rightarrow B$ is a vector bundle and $B$ is paracompact, there is a continuous function $\mu : E \rightarrow \mathbb{R}$ such that on every fiber $\pi^{-1}(x)$, $\mu$ is a positive-definite quadratic form.
Is there a nonexample when $B$ is not paracompact? I don't have any strategy to prove that no such $\mu$ can exist.