Do all vectors have direction and magnitude? 

I go by Vector. It's a mathematical term, represented by an arrow with both direction and magnitude. Vector! That's me, because I commit crimes with both direction and magnitude. Oh yeah!

For Euclidean vectors, it's natural to think of vectors having direction and magnitude.  What about vectors in abstract vector spaces?  Is it possible to (sensibly) define notions of direction and magnitude for all vectors?
(If not, what is an example which highlights why it's not always possible?)
 A: Summary
There's a reasonable definition of direction in abstract vector spaces, but that doesn't always include "orientation." To talk about magnitudes as lengths, you really need extra structure provided by a norm into an ordered field.
Direction
In abstract vector spaces, you can link a weak idea of "direction" with a vector directly by just defining:

We say $v$ and $w$ have the same direction if $\langle v\rangle=\langle w\rangle$. ($\langle\dots \rangle$ is denoting linear span, here.)

That is, each $1$-dimensional subspace could be thought of as a class of vectors in the same direction. Notice, though, that this scheme has you think of the zero vector as being "in all directions," and maybe philosophically then it has no direction at all :)
Some folks might also include a component of orientation when they're thinking about "direction," so we should discuss that too. As far as I can tell, this necessitates $F$ to be an ordered field $F$ so that you can establish a dichotomy of what is positive and what is negative. (You don't have to have orientation if you're happy with the definition of direction above, but I think it's worth discussing.)
Given a direction $\langle v \rangle$ (and an ordering on your field of course), one could say that the elements of $A=\{\lambda v\mid \lambda>0\}$ are mutually oriented in the same way, and $B=\{\lambda v\mid \lambda<0\}$ as oriented in "the other" way, and they are oriented oppositely to those things in $A$. This is a problem for finite fields, which can't be ordered. In characteristic $2$ fields, for example, $v=-v$ for all vectors, so the dichotomy doesn't exist at all, there.
Magnitude
To talk about lengths in an abstract vetor space, you really need an extra structure called a norm. This gives you a way to measure how "long" vectors are. Being able to compare lengths of vectors with a norm again only makes sense when you are working with a norm into an ordered field, so that you can distinguish which magnitude is greater.
On the other hand, if you're just happy to have some sort of scalar for each vector, then there are generalizations of norms into nonordered fields that would work. You just couldn't interpret their values as lengths. Really, our geometric intuition about length is all bound up in ordered fields.
