First, make sure that you have loaded data that can be edited in ArcMap. If you load data that cannot be edited, such as images, the start of editing will be grayed out and cannot be used. If shp is still grayed out, it means that there is something wrong with arcmap, and it is better to reinstall it, because this kind of problem is rare and it is difficult to find out the reason. I have been using arcgis for many years and have never encountered such a situation.