Can vectors even be expressed unambiguously? Vectors are abstract concepts. Lets take one of the simplest, more concrete vectors out there: an euclidian vector in 2D. Now, I think that even such a vector is abstract, it cannot be written down, it cannot be expressed, it cannot be specified, it cannot be conveyed. At best you can give it a name, like $V$.
What one could try to do, is to express it as a linear combination of other vectors of that space, for example an orthonormal basis $\{B_{1}$, $B_{2}\}$.
For example, it may be that "$V = 28\cdot B_{1} + 7\cdot B_{2}$"
Or, "$V = (28,7)$ in the basis $\{B_{1}, B_{2}\}$."
The problem is that I expressed the vector in terms of other vectors. $(28,7)$ means nothing unless I can somehow describe $B_{1}$ and $B_{2}$. After all, if I chose another basis $\{C_{1}, C_{2}$, $(28,7)$ would represent a completely different vector.
And I can't describe $B_{1}$ or $B_{2}$, express them, other than by doing so in terms of other vectors, just like I couldn't do it for $V$.
So I cannot specify which vector $V$ is, other than adding two new vectors, which I also can't specify. It all seems completely circular to me.
Trying to frame this as a question: how can one write down a vector in a way that it actually specifies which vector it is? How can someone even specify what basis he is using? Aren't all those expressions circular and meaningless?
 A: Vectors and vector spaces are abstract concepts it's true, such as the concept of a group or a field. But that doesn't mean we can't work with these objects in a concrete setting.  
When you are working with a vector space such as "a Euclidean vector in 2D", you are usually specifically working with a coordinate system, really in this case you are specifying a lot: you have a measure of distance, angles, etc. (this is structure given by the "Euclidean" requirement. In this case you have a standard basis that you are using by default, you are representing your vectors in $\mathbb{R}^{2}$ with orthonormal basis vectors $(1,0)$ and $(0,1)$. You can then specify an alternate basis to use in terms of these vectors, if you want (the computation of distances and angles in terms of coordinates can change depending on the properties of your basis). There is really no meaningful ambiguity here.
Abstractly, we can talk about having a 2-dimensional vector space over $\mathbb{R}$; here we are not specifically saying what the vectors look like, we could be wanting to represent them as polynomials, or maybe we don't even want to say. Here we work in terms of showing things about the vectors that don't depend on what they look like. If you and I have a different basis, the results we are proving just depend on the representation; our points of view are equivalent, so it doesn't matter if there is some ambiguity.
A: Vectors are not abstract concepts. Vector fields for example are very common in nature-one is produced by your monitor as you read these lines. 
Using a magnet and iron fillings you can see such a field yourself, like this one which is just a collection of vectors:
Now as for vectors being used to define other vectors creating a paradox..
Well, we use numbers to define other numbers all the time in mathematics but the nature of what is a number is largely left out of the discussion.
A vector representation changes from basis to basis but the vector itself does not. If this creates a degree of confusion consider the analogy of representing numbers in terms of different basis. Base $10$ gives as the decimal system but base $2$ gives as the binary system. Base $54764673$ gives as another-a not really practical one.
As an example, $9=9_{10}$ on the decimal system equals $1001_{2}$ on the binary.
$9=9_{10}=1001_{2}$
They look very different as two vectors represented on two different basis look, but they are the same number, the quantity $9$. Different representation of vectors preserves the properties of the vectors the same way different numeral representation preserves the quantity the number symbolizes.
Besides, first you define a system of coordinates axiomatically (the vectors that form the basis).
Also the multiplication of a vector with a scalar is what makes the representation work. And that can be reduced to multiplying a line segment with a number producing another line segment, a concept easily grasped.  
