Since the water shut off your valve must have the "valve closing" threads near the valve sealing point deep inside the valve, not out near the packing nut you've removed. The stem broke off from the "head" of the valve with the threads and seal.
In theory if you could get the water shut off upstream (might mean the main shutoff, evidently, and you'll have to do that for ANY repair option) and get the valve head out (difficult, as it's deep inside with a broken stem) and find a replacement complete valve stem assembly for this exact sillcock, you could just replace the valve stem assembly.
In practice, that might be well-nigh impossible at several points, and ripping the finished wall open to access the inner end of this assembly so you can replace the whole thing is much more of a sure bet for solving the problem.
If most of the pipes you can see are soldered, you will almost certainly need to unsolder this to replace it. If you have threaded pipes for your plumbing you may be able to "just unscrew it" or it may be firmly stuck and require opening the inside wall to apply a wrench to the joint. Trying to unscrew it from the outside may twist and collapse the outer tube, which appears to be copper. A brass one is commonly less likely to collapse. Trying to "unscrew" a joint which is actually soldered will almost always twist and collapse the tube/pipe.