In a comment, John Lawler wrote:

> (1) _What to do?_ is not a sentence; there's no subject, for one thing. (2) The construction is not particularly archaic, although _wh_-infinitives have been around for a long time. They're still widely used. What they are isn't so much archaic or formal as it is **formulaic**. That is, everything is taken for granted and/or given in the context, so out of context they make no sense.

It may not be immediately clear to everyone why John said that there’s no subject in *What to do?*. I imagine some people are thinking that the *wh-* word there is the subject, but it’s not. It’s actually the object of that infinitive clause. Imagine a book title like *Whom to call?* There it’s clearly acting as the object of the verb *call*. If the answer is *him*.

* I don't know whom to call.
* Whom to call? 
* Him.
* I’ll call him.

So it’s an object there.

That’s not to say that infinitive clauses cannot have subjects; they can do so, but this is not one of those cases. Examples of infinitive clause with an actual subject is:

* For her to call him that night was all he had ever hoped for.
* I need her to call me.

Just as not all sentences are questions, not all questions are sentences. That doesn’t make them ungrammatical. They simply aren’t sentences, since all sentences have a subject and a finite verb. When you have a *wh-* word plus an infinitive ending in an utterance ending with a question mark, you have a question. Any utterance that ends with a question mark is a question: it’s one inviting an answer. And not all answers are sentences either. They’re still answers.

> Question: Ready?

> Answer: Ready.