In the absence of the Axiom of Choice, it is not necessarily true that all vector spaces over a field have bases.

What are the possible global dimensions of fields in a model of ZF in which AoC does not hold?

For comparison, a related result of Osofsky is that the global dimension of a countable product of fields is $k+1$ if $2^{\aleph_0}$ is $\aleph_k$, so that dimension depends on how badly the Continuum Hypothesis fails in one's model of set theory.