I wonder if they can install the barrel to the same exact specs as the one I have on currently...
NO. Not EXACTLY. First off, the one you have on now is used, "worn in" to some degree by the rounds already fired. EXACTLY matching it would require painstaking effort, AND the original barrel to measure from.
What you will have with a new barrel (properly) installed, is a new barrel, headspaced to pass the GO gage, and not pass the NO GO and Field gages. Beyond that, how closely it matches the precise dimensions of your old barrel is a matter of serendipity. It will pass SAAMI specs, everything else is a matter of luck of the draw.
Unless the barrel is used in rapid fire, .308s usually go a good 3-5K rounds before the accuracy of the barrel starts to seriously fall off. Some go a good bit more than that.
When a barrel is "shot out" is a matter of your judgment, and requirements. If you need sub MOA, a barrel that is worn to the point it only gives 1.5MOA is due for replacement. A barrel that gives 1.5MOA is still a good barrel if you need "minute of deer", and doesn't need replacement. So, what are your needs?
Also, be aware that even the finest replacement barrels may not shoot as well as the original. Not highly likely but it does happen. The main point to having to work up new loads for the new barrel (besides basic safety) is to get it to shoot as good as, or better than the original. Usually possible, but not always.