Calculate distance covered by points in a shapefile

Can you type a value into the Distance field, or will it not let you? Are the points that are supposed to be connected spaced out mostly uniformly, and do you know that distance? You could use just over half that inter-point distance (50.1%) to set the distance for the buffer, to keep it as small as possible while still making the buffer overlap. From your picture, it looks like most of the points are aligned vertically and horizontally, but if you have any diagonally connected points, you might consider a larger buffer distance than that.

If distance is mostly uniform, you also could multiply the number of points times the distance between them to get the distance covered. This would be a slight overestimate, since every endpoint technically should not count toward the total, but it's a fast and painless way to get a rough answer. I'm not sure how much precision you need.

If you have access to the Spatial Analyst extension, you could try:

1. Euclidean Distance on your points, set an appropriate “max distance” if desired. http://desktop.arcgis.com/en/arcmap/10.3/tools/spatial-analyst-toolbox/euclidean-distance.htm

2. Use the “Con” tool or “Raster Calculator” to mask appropriate zones. I.e. Con(EucDistance <= 125, 1)

3. Use the “Thin” tool to skeletonize these zones. http://desktop.arcgis.com/en/arcmap/10.3/tools/spatial-analyst-toolbox/thin.htm

4. Convert raster to polyline if necessary.