Of course, the sentence "
No!" may be used to signify:
That it is a sentence can be seen from this: it might be the only word ever spoken to us by someone, even someone that we had never ourselves spoken to -- and yet, given the right circumstances, we would understand them perfectly.
This is a reasonable test of sentencehood: whether, isolated like this, it can be understood. Often this is a case of chasing dangling parts of speech, as in the grammar-school idea of a sentence given below by Phinslit, but this task is subservient to the idea of semantic completeness. Sometimes, understanding can occur without the kind of specificity required by the grammarians.
Regards the corresponding question of whether 'Yes' must always imply some propositional scaffolding in order to make sense, a close reading of the final chapter of Ulysses is advised.