Question: Is there a set of sentences in the language of ordered fields whose models are precisely the rationals and any ordered field isomorphic to them?

Answer: 
The answer is no. For by the) [Lowenheim-Skolem theorem,](http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem) if the set $T$ of statements has an infinite model, then $T$ has models of arbitrarily high cardinality.
The existence of such a set of sentences is precluded by the compactness theorem of first-order logic. Consider the sentences
\begin{align}
a & > 1 \\
a+a & > 1 \\
a+a+a & > 1 \\
a+a+a+a & > 1 \\
& \,\,\, \vdots
\end{align}
Every finite subset of this set is satisfiable within the rationals. Therefore by the compactness theorem, all of them are simultaneously satisfiable in a structure satisfying all first-order sentences true in $\mathbb Q$. But such a structure clearly cannot be isomorphic to $\mathbb Q$.