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I know what the span of a set of vectors means, but sometimes we say a set "spans" another. These questions are about the specifics regarding the latter usage.

Question 1: Can we say that a set of vectors spans another set of vectors if that second set of vectors isn't a vector space?

Question 2:

Can a set of vectors $S$ be said to span a vector space $V$ if $Span(S)$ is larger than $V$? For example, does $\left\{\begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}, \begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}\right\}$ span the set of all vectors in the form $\begin{bmatrix} 0\\ a\\ b \end{bmatrix}$ for some $a, b\in\mathbb{R}$?

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    The answer to question $2$ is yes: *The span of $S$ has to contain $V$*, and the word "larger" cannot be used in this context (what does it mean for a set to be larger than another, if both are infinite, for example?). As for question 1: not true.A set of vectors $S$ spans $V$ if $V \subset Span(S)$. There's nothing more to be said: $V$ could a vector space, infinite dimensional, large set, but it can still be spanned by some other set $S$. – 2017-02-07
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    Have a look at this page: https://en.wikipedia.org/wiki/Linear_span – 2017-02-07

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As to your first question, the span of vectors is always a vector space, so it doesn't make sense to write that a set of vectors span something other than a vector space.

Now, as to your second question, yes. Think of it, what it means to span is that every vector in the space can be represented as a linear combination of vectors in the space. Thus the span of all the basis vectors also span all of the subspaces up to the space that you are interested in. I.e you could take a subset of your space that would span a subspace of V.