Are you talking about a Savage model 110? There would be some issues you need to over come to shoot a 22 hornet in a Long action Savage or a short action Savage for that matter. Yes you can have a barrel built for the 22 Hornet on the Savage action. The problems you will run into are a few. First the head size does not match any cartridge that the Savage is chambered for. To get it to work, i.e. extract and eject, you will need a custom bolt head made or a standard one modified to fit the head of the Hornet case. The back of the barrel will have to be made in a cone shape so an extractor will slip over the case head. That is the way my Model 40 (22 Hornet) works. You will also need to do some thing about the magazine. A 22 Hornet will not feed correctly in any magazine that Savage makes that is in a 110 series or a 10 series. You could make it a single shot and feed one at a time.
I have a few Savages that I have swapped barrels on. I would not recommend doing it very often. When I do, I remove the scope, and mounts, take off the old barrel and replace with a new barrel. I then reset the mounts and scope and rezero the scope. You would have to swap out the bolt head or a whole bolt assembly before setting a different barrel. Not something I want to do often.
If you really have your heart set on a 22 Hornet, Savage makes a model 25 that is chambered in 22 Hornet. There are several other rifles that are chambered in 22 Hornet as well. i have a Savage model 40 that is a single shot 22 Hornet and there are other Savages that have been chambered for the 22 Hornet as well such as the model 340 that are no longer in production, but can be picked on the used market.
I have put a 221 Fireball on a Savage short action that is like a souped up 22 Hornet. The head and body diameter is the same as a 223 so you can use a 223 bolt head and make it a single shot and they work great. I never tried to feed the 221 through a 223 magazine, but I think it should work fine.
BTW the 221 Fireball is a great varmint round out to 200 yards or a bit further. They handle heavier bullets much better than the 22 Hornet. 50 grain bullets are a standard for them, but you can shoot heavier ones if you get a twist rate of 1-12 or faster. It is only a step behind the 222 Remington and about 300 fps slower than the 223. I have the 22 Hornet, a 22K Hornet and 221 Fireball. If I had to pick between them, it would be the 221 Fireball with the 22K Hornet a close second. The case life on a 22 Hornet can be short. The case life on the other two is much longer. You can also load a 223 down to 22 Hornet levels too, as another option.
Good Luck and Good Shooting