A vector space $V$ over the field $F$ can be more formally described as a set with some properties. These properties include closure under addition, and closure under multiplication with a scalar in the field $F$.
Many concepts can be thought of as vector spaces. Whether it is useful to think of them as vector spaces is another question.
Here are some examples:
The set of real numbers is an infinite dimensional vector space over the field of rational numbers.
The set of real numbers is a 1 dimensional vector space over the field of real numbers.
The set of rational numbers is a 1 dimensional vector space over the field of rational numbers. It is not a vector space over the field of real numbers because it is not closed under scalar multiplication. It is not a vector space over integers because integers does not make a field.
If we take the co-domain to be the set of real numbers or the set of complex numbers:
The set of functions over a set domain is a vector space over both the real and rational numbers.
The set of bounded functions over a set domain is a vector space ...
The set of [insert description here] functions is a vector space over the real numbers/rational numbers that you can add any two such functions, or multiply one of these functions with a real/rational and still get a function that fits the description. For example: continuous, uniformly continuous, integrable, $C_n$, polynomials, rational functions, etc.
The set of magic squares is a vector space (I believe 4 dimensional) over the real numbers. It is also a vector space over rational numbers
The set of equations is a vector space.
Ultimately, you can come up with some really silly example of vector spaces if you use some really unfamiliar/obscure fields. For example, this field.