It will depend on the shrub as to how far out you need to dig. You want to get the ball and roots without disturbing if you are going to transplant it. For removal only, you don't have to be so careful but you will want to get as much of the root system as possible or it may grow back.
Dig in a circle at least 18" from the base of the shrub on all sided then continue to work deeper with the shovel cutting thru roots as you go. It will stunt the shrub until it develops new roots if you are replanting it. If not don't worry about the roots and just dig it out and get as much of the root as you can. If it is a tall shrub and you just want it gone, cut it to about a foot high so it is easier to manage.
Digging out a shrub is hard work. The easiest way is to tie a rope to it and pull it out with a tracker to vehicle.
Hello are You removing the shrub to discard or to transplant? Transplanting obviously requires more care.