It turns out that, if you restrict to spaces of finite dimension, then Euclidean space is the only vector space out there, up to your choice of labelling the basis elements and up to a certain linear transformation.

In this sense, vectors can always be written as lists of numbers, but no, they don't have to be this in general. For example, you could have a vector space of smooth functions. It's infinite dimensional, and its elements (its vectors) are functions, not tuples of numbers.