How small a location is possible to identify when any visible star is at its zenith directly above? How small an area on Earth is it possible to identify with any precision when any celestial object (star, planet, comet, nova) (visible with the naked eye alone) is at its zenith directly overhead?
--Would it be possible to identify an area as small as a town, or village, a square mile, or an acre, or even a building? 
--How big an earthly location would be required to narrow it down if such a celestial object is directly above?
  Thanks.
PS:  I've tried to narrow the question down to get a simple to understand answer.  I'm not very technical, nor an astronomer.
Lets assume your observer has a good watch, so she knows the exact UTC time. But lets also assume that she has no special kit, and has no special training. 

She might be able to estimate the location of "directly up" to an accuracy of about one hand width (at arms length) which corresponds to roughly 5 degrees.  To achieve better than this, she would need some tools for sighting the position of stars.

With an error of +- 5 degrees, the error on the Earth's surface of any calcuation based on her observation is also 5 degrees of the Earth's surface. That is roughly a circle of radius 550km (350 miles).  

Such an observation could prove you were in Mexico and not Canada. But it couldn't distinguish Conneticut and Washington DC, much less a square mile.

To get 1 mile accuracy you would need measurements at least 350 times better, This is better than one-sixtieth of a degree. The most skilled observers did achieve this level of accuracy with naked eye equipment, but not from a single observation. It could be done with a telescope designed for measuring the position of stars, provided the operator was sufficiently highly skilled.

If your observer doesn't have a good watch, she is subject to the problem of Longitude: It is relatively easy to find Latitude just from astronomical observations. It is much harder to find the longitude.