How can I estimate the miles of railroad track from an image?

I suggest checking out OpenStreetMap or TIGER line files. Both have the information you need; just make sure you consider what level of detail you're looking for.

You don’t really have to estimate; the data for railroad tracks is often available online as shapefiles or KML. You might want to check those sources first to get accurate information.

One way to estimate it is to count the red pixels in your image, divide that by the track width in pixels, and then calibrate the miles based on a known distance between two points on the map, like Seattle to Miami.