Usually in mathematics one defines these structures as tuples
A field is a triple $(K,+,\cdot)$ such that $K$ is a set and [...] and $\cdot:K \times K \rightarrow K$
A Vectorspace is a triple $(V,+,\cdot)$ such that $V$ is a set and [...] and $\cdot: K \times V \rightarrow V$
So your question is meaningless: A set (say $\mathbb R^2$) cannot be a field or a vectorspace or a group or anything - only if you add some additional structure (most of the time operations) you can ask this question.
For example $\mathbb R^2$ can be the set used in the definition of a field, as well as the underlying set used in the definition of a vectorspace. And we are happy, the addition operation $ +:\mathbb R^2 \times \mathbb R^2 \rightarrow \mathbb R^2 $ is the same, and the multiplicaton for "the" vectorspace structure $ \mathbb R \times \mathbb R^2 \rightarrow \mathbb R^2 $ is "compatible" with the multiplication for "the" field structure $ \mathbb R^2 \times \mathbb R^2 \rightarrow \mathbb R^2 $