Minimum Bounding Geometry Irregular Polygon Length & Width
Hi Ken Walsh , not sure where to start, since there is a lot going on here.
I notice that you have a metric coordinate system and the width and height of the area is very small 35 - 40 cm. This affects the data. What does the data represent?
I notice that you send me a polyline shapefile and not polygons. When you want to obtain polygons digitizing them from a background (the image) you should directly capture them as polygons to avoid problems. Better yet, as the background image seems to have a continuous range of values, you could probably use a threshold value to create a raster containing only the areas you are interested in and convert those those raster areas to polygons (no need to digitize the polygons).
What I did with your data was to "move" them to a different location and give them a bigger size by defining the WGS1984 geographic coordinate system and projecting them to Web Mercator Auxiliary Sphere (which obviously will not be correct without knowing what the data represents). When I used the Feature to Polygon tool, it just created polygons for a small part of your data (see the green polygons in the image below). You can use the MBG (area by rectangle) on the lines as well to create rectangles around the lines (see purple-ish color rectangles).
In case the polygons don't have too many curves, you could approximately determine the width of the polygons, by using the length from the rectangles and divide the area of the polygons by the length. This will approximately represent the average with of the polygons, but bear in mind that when a polygon has more curves the length is too small and the with will be bigger than it should be.
Thanks Xander, I will admit that I don't have very much GIS experience so some of what you detailed I may have to ask you about. The background is a layer of clay-cake, not very large as you pointed out. This is a geology experiment. When you apply tension to the cake from the left and right, small cracks will start to form, these represent rocks faulting. As more pressure is applied these small cracks grow into each other creating larger longer faults. So I have a selection of time series tiffs representing different strain amounts. I created a shape-file with a circle, an area of interest, see attached. I don't want near the perimeters as they will start to bow as the stretching increases. What I want to get from each of the cracks is the mid-line length and maximum width, if possible.
Now knowing GIS, I managed to get the tiffs in with a worldfile that I think makes sense. Overlay the tiff with AOI circle and then create a new shape-file and start by hand plotting each of the cracks. I had to choose a coordinate system so I chose WGS1984_UTM_Zone_13, I have no idea if thats what I should have chosen as this is a tiny rectangle not representing anywhere. To map each slide, as you can imagine, takes hours and there are a lot of slides, so if there is a more efficient way to accomplish this I would love to learn it. I admit from "threshold .... Feature to Polygon" in your post I was lost; ignorance is a curse. There are no curved lines on the polygons. Hope this makes it a little clearer.
The "minimum area bounding rectangle (MABR)", as Xander has demonstrated, is the route to go.
Sadly, continuing to digitize in the interim would be the route to go, unless you want to venture into some form of image classification.
It appears that you will also have some issues with this given the lighting on the image not providing a unique signature across the image.
As a shortcut, you might want to just consider digitizing a rectangle of the approximate width and length, rotated appropriately over the image cracks. The finely digitized boundaries are going to get reduced to a rectangle anyway.... and I suspect "human-ware" might do a better job representing the aligned rectangle.
If you find a persistent mid-axis width being substantially larger, then a 'selection' of those features could be examined with the "minimum area bounding ellipse (MABE)" The MABE should fit nicely in the MABR if sufficient perimeter points are obtained. The MABE can be sliced to get widths across the long axis if needed... but you aren't going to do hundreds at once in any event, so stick with the MABR for the majority of work and look at the MABE for longer cracks.
Great application of GIS technology BTW. 
