Stainless steels:
Stainless steel is chosen for its corrosion and rust resistance. The grade of stainless steel most commonly used for firearms barrels is AISI #416, which is a free machining variation of #410. The main constituent in this stainless steel is chromium. Chromium improves wear resistance, and corrosion and scaling at elevated temperatures. The reason this is important for firearms, especially target rifles, is the wear that degrades accuracy is the hot, burning powder gasses eroding the throat area. The throat is the spot in the barrel where the bullet enters the rifling. Without a good start to the bullet in the bore, the gun won't be accurate. If you're going to have a barrel and expect it to last for 20,000 rounds (target rifle), you'll have to select a material that will last that long. The attributes that make stainless steel so desirable for a firearm is unnecessary in an air gun because there are no hot, burning powder gasses. The other feature that most people look to stainless steel for is rust resistance. They use the term "resistance", not "proof", because the grade of stainless steel used for rifle barrels can rust. Stainless steel barrels are not maintenance free. Just because you have a stainless steel barrel doesn't mean that you don't have to take care of it. So it would be a mistake to put a stainless steel barrel on an air gun in the belief that you can now neglect it. At the moment, I don't see where the extra expense and difficulty of machining and rifling a stainless steel barrel would be an advantage. Without the rest of the rifle's components being made of stainless steel, and having a synthetic, weather proof stock, the stainless steel barrel only is no advantage.
THE STEEL.
The barrel of any firearm is a pressure vessel with the action serving to stop up the rear end. The peak pressures involved during discharge are enormous, 50,000 pounds per square inch or more, and special steels are required to safely withstand these stresses.
Two classes of steel are currently used in rifle barrels. Most barrels for use on hunting rifles and in military firearms are made from a high alloy Chrome Molybdenum steel of the sort used in high stress components such as truck axles, conrods and such. In the United States these steels are designated as 4140, 4150 and 4340 types. In Britain these steels are better known as EN 19 or EN 24.
In target shooting stainless barrels have for the most part supplanted the use of Chrome Moly barrels. The steel is not a true, fully austenitic stainless such as is found in cutlery. The 416 type stainless steel used in barrels is one of a group of martensitic steels which can be hardened by heat treating like regular Carbon steels. 416 stainless is more accurately described as a "free machining, rust resistant" steel having a high Chrome content, around 10%, but with sulphur added to give it good machining qualities. It is widely considered that stainless barrels will have a longer life and are more accurate than Chrome Moly barrels. If stainless barrels are "shot in" using the prescribed procedure, the barrel aquires a burnishing which almost eliminates fouling, so making stainless barrels very easy to clean.
Because stainless steel is more expensive than Chrome Moly steel and it is more difficult to black due to the Chrome content, high production makers of hunting and military rifles prefer to use Chrome Moly steels. But target shooters who want the best possible accuracy from their barrels are almost without exception choosing "stainless" barrels these days.
The tensile strength of the steel is measured as the force required to break a rod of steel having a one inch cross sectional area by pulling it from its ends. The tensile strengths of steels used for making barrels should exceed 100,000 lb per square inch giving at least a factor of two safety margin over the chamber pressures experienced during firing. But the impact strength of the steel is probably even more important, this being the ability of the steel to withstand a sharp knock without breaking. Generally speaking, the tensile strength of a steel can be increased by hardening it. But as the hardness is increased, so the steel becomes more brittle and it becomes more susceptible to fracturing from a hard knock or sharp impact - or setting off a small explosion inside a tube of the stuff! A trade off must therefore be made of tensile strength against impact strength and for barrel steel the resultant hardness settled on is usually between 25 and 32 on the Rockwell C scale.
The heat treatment and other production processes involved in making the steel bar leave residual stresses, which can result in the bar bending as steel is removed in making the barrel. The stress can be relieved by putting the steel in an oven and taking it up to 600 C, then allowing it to cool very slowly over the next twelve hours or so. Barrel steel is usually double stress relieved to make absolutely sure it stays straight through the various machining processes.