Why are vector spaces defined over a field? I have a general intuition of what vectors are/look like in the context of physics and what not. But it doesn't seem too closely related to properties of a field. So why are they defined over a field and not some arbitrary ring or something. Might be missing something...

They are defined over general rings, but vector spaces all have a basis, while modules need not have one. Modules which have a basis are known as free modules (of finite type  if they have a finite basis).
Worse, if the ring is not commutative, it may happen a free module has bases with different cardinalities.
Most modules have no basis. The simplest example would be the ideal $(X,Y)\subset K[X,Y]$ $\;K$ a field), which has $\{X,Y\}$ as a minimal set of generators, but they're not linearly independent since $\;Y\cdot X-X\cdot Y=0$.
