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.