set theory - If V=L implies the axiom of choice, why can't we construct the things that depend on the axiom of choice?

$V=L$ is a statement in set theory asserting that every set is constructible. It implies the axiom of choice.

I find this kind of confusing, since the axiom of choice is non-constructive. Can you, for example, find/construct a specific basis of any vector space in $ZFC + V=L$, or within $L$ itself?