Knowing the resolution helps. If you have access to Python and GDAL, you can do something like (not run):
from osgeo import gdal
# Read the original file
fn = "c:/fullpath/myfile.tif"
ds = gdal.Open(fn)
# Write out a copy, changing the GeoTransform
driver = gdal.GetDriverByName("GTIFF")
ds_out = driver.CreateCopy("c:/fullpath/out.tif", ds, 0)
ds_out.SetGeoTransform([0, ds.RasterXSize / 2400, 0, 0, ds.RasterYSize / 2400])
ds_out = None
ds = None
The key part is the GeoTransform, it is
[xorigin, xpixelsize, 0, yorigin, 0, ypixelsize].
Having the proper origin (0,0) in the upper-left, and a physical pixel size (in inches), you will be able to make relatively accurate measurements. You don't need to set a projection when opening it ArcGIS. It will show as unknown units, but you know that they are in inches. Having a ruler or other known measure in the picture would help confirm that everything is as it should be.
The simpler alternative is to do any tracing / analysis in 1x1 unit pixels, as it is in the raw image, then convert distances and areas appropriately using the known resolution (1 pixel = 1/2400 inches, for distances, in your case).
If you need to assign a projection, something like this
"+proj=eqc +ellps=sphere +units=int-in" might work:
PROJ.4 : '+proj=eqc +lat_ts=0 +lat_0=0 +lon_0=0 +x_0=0 +y_0=0 +a=6370997 +b=6370997 +units=m +no_defs '
OGC WKT :
GEOGCS["Normal Sphere (r=6370997)",
I don't think it really matters what you pick as long as the units and pixel size are correct, because you are dealing with such a small area, geographically speaking.