Almost by definition, there isn’t a standard size or weight. A sawed off shotgun is just an illegally modified weapon, that’s been cut down to make it easier to conceal.
In general, civilian pump action shotguns tend to be roughly 6.5 to 7 lbs. Barrel length varies depending on what the shotgun is intended for. Hunting shotguns frequently have ~28” barrels, combat versions frequently have ~18” ones or shorter.
The weapon itself can easily add another 20” inches to the overall length, so you’ll end up with a 38” firearm. (I’m using the Remington 870 as an example if anyone’s curious.) At slightly over three feet, you could reasonably stuff it under a full trench and go on with your day.
With a shotgun, the barrel’s length doesn’t affect it’s damage, but it does affect the spread. Longer barrels will have a narrower spread, and be effective at greater ranges.
Cutting the barrel down results in a wider spread and a shorter range weapon, in addition to making it easier to conceal.
With a pump-action (or tube fed semi-auto) shotgun, you can’t gut the barrel shorter than the magazine (the second tube running under the barrel). So, that’s a hard limit, and it will vary even with specific shotgun models.
The Remington 870 I mentioned above has varying magazine capacities ranging from 4 to 7 shells. Doing some quick math; a 12 gauge shotgun shell is between 2.5 and 2.75 inches, so, your 7 shell magazine could be nearly 20 inches long, meaning you couldn’t cut enough off a 20 inch barrel for it to matter.
The second thing that can be cut is the stock. This will reduce your ability to properly stabilize the shotgun, but, again, it will make it easier to conceal. Ignoring bullpups, I’m not aware of any shotguns that actually run vital components into the stock, but that is an issue with some rifles (including the entire M16/M4 family), so if you’re starting with some exotic full auto shotgun, getting rid of the stock might cause problems.
With a breach loading shotgun, you can cut it down to an almost pistol sized weapon. Anecdotally, these will have a very wide spread, though I couldn’t tell you what their effective range is or what they weigh. You might be able to stash one of these in a custom shoulder holster, and it might be short enough to hide under a jacket.
Small military/law enforcement shotguns for use in close quarters exist, and these can end up with 10” barrels. Combined with the pistol grip, you can end up with a shotgun that’s under 20” long. This won’t quite fit under a jacket, but a thigh length coat should be able to conceal it.
I don’t know the weight for those, but at a guess, I’d stick it at around 5 to 6lbs.
The primary use for sawed off shotguns is going to be indoors in cramped spaces, where a full length weapon would get in the way or be awkward to use. Homes and apartments are the primary examples. Existing compact shotguns are used primarily as breaching weapons, to destroy a door’s latch and hinges with a frangible round before entry.
It might also be worth it to look at this post from July. That was about a near future urban combat environment, but some of the details might be applicable, so long as you remember the specific weapon names I was picking were more for giving them high tech looking weapons rather than actual combat workhorses. (I’m specifically linking to my response to ProRonin because I forgot to talk about carbines in the original post, not for the powered armor.)