This is partly redundant with previous answers: one can present the students with a vector space that does not have a natural basis. What I would like to stress is that the simplest is probably the best, at least for a first example, and that the simplest is to take a vectorial plane in $\mathbb{R}^3$. What would be the two coordinates of a vector in $V=\{(x,y,z) | x+y+z=0 \}$?
I confess that students could be trying to think of these vectors in $\mathbb{R}^3$ rather than in $V$, so that this example maybe better to explain the need of a definition for a basis.