you are confusing vector space with 3-dimensional physical space. while 3-dimensional physical space can be modelled by a vector space (and indeed, it helps to visualize certain concepts such as orthogonal, and length of a vector by imagining things that might exist in our 3-dimensional world), the concept of vector space is MUCH more versatile than just describing physical situations.

for example, the set of all real polynomials of degree 3 or less, is certainly a valid set of things to contemplate. we can write such a polynomial as:

, which we can think of as a 4-vector (a,b,c,d).

this set obeys all the axioms of a vector space over , and i claim that {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)} is a basis.

now by this, i mean the 4 polynomials:

,

it is clear that span the set of all polynomials of degree 3 or less, just use the coefficients as the scalar multiples for the linear combination of basis elements.

suppose was the constant 0-function: 0(x) = 0, for all x (a horizontal line for its graph: the x-axis).

then p(0) = a, so a = 0. so . since this is always 0, for all x, and x is not 0 for any x BUT 0, it must be the case for every x except 0, that:

, for every x, except possibly 0. but surely such a function is continuous, so it has to be 0 at x = 0 as well, by continuity.

and THAT means that q(0) = b = 0, so .

again, this means that c+dx = 0 for every x except possibly 0, so from x = -1, we see that:

c - d = 0, so c = d.

and from x = 1, we have:

0 = c + d = 2c, so c = d = 0.

so the only polynomial p(x), of degree 3 or less, which is 0 for ALL x, is the polynomial:

, which proves that is linearly independent, and is thus a basis.

(of course it's easy to see that if a(1,0,0,0) + b(0,1,0,0) + c(0,0,1,0) + d(0,0,0,1) = (0,0,0,0), we must have a = b = c = d = 0, but i deliberately wanted to show this another way, directly from facts you know about polynomials).

so there's one vector space which definitely has a basis of 4 elements. in fact, there's no reason to stop at 4, we can have vector spaces with 5,6,7 or any positive integer dimension, in fact there are even vector spaces with infinite bases.

now, you might think that such things are just "abstract" and don't have anything to do with "real-life stuff". but think of linear transformations in 3-dimensional space, such as rotation, or reflection (these are behaviors which real things exhibit, right?). these can be described by 3x3 matrices, and the set of all real 3x3 matrices is a vector space with 9 dimensions.