Q:

Geodesic measurements for short distances throughout US?

My ArcGIS experience has been limited to working on large-scale maps with planar projections. I'm moving to a project in which I want to calculate the distance between pro sports facilities throughout the U.S. and their city's respective downtown/central business district (both defined as geocoded points).
My primary interest is in relative distances, and my question is whether using a geographic coordinate system for US map points and using "Near" to calculate geodesic distances between stadia and downtowns is a sound approach to avoiding issues that would arise from calculating distance based on a projection of the US?

A:

I am afraid that the answer is no: what you mean to do is correct but it will not work like this in ArcGIS with Near. 
From the help, you can read that :

The distances calculated by this tool are in the unit of the
  coordinate system of the input features. If your input is in a
  geographic coordinate system and you want output distances to be
  measured in a linear unit (as opposed to decimal degrees), you must
  first project your input to a projected coordinate system using the
  Project tool. For best results, use an equidistant projection or a
  projection intended for your study area (UTM, for example)

in other words, if you use a geographic coordinate system, near will work as if it was a cartesian coordinate system. 
You should instead use the Haversine approximation (see here ) or, better, the Vicenty's formula to find your solution (see @Michalis Avraam's answer)

