You could try giving the following example: the set of all positive real numbers, considered as a vector space over the field R, with vector addition given by multiplication and scalar multiplication given by taking exponents.
As a first step, you could verify that this satisfies a few of the vector-space axioms, and then let students check the rest of them (say, as homework). Then, you could ask questions like, "what is the dimension of this vector space?" or, "give an example of a (nontrivial) linear transformation from this space into R^3."
