From Google
As a general rule, carbon steel is tougher and harder than most stainless steel. This is primarily due to the higher carbon content in carbon steel.
However, it's important to consider:
Varieties of Steel: Both stainless steel and carbon steel have different grades with varying properties. For example, some martensitic stainless steels can be heat treated to become very hard.
Ductility vs. Hardness: While carbon steel is generally harder, stainless steel often exhibits greater ductility, meaning it can deform without fracturing.
Applications: The choice between carbon and stainless steel depends on the specific application. Carbon steel is often preferred for applications requiring high strength and hardness, like cutting tools. Stainless steel, with its corrosion resistance, is suitable for applications where preventing rust is crucial.
In summary, while there are exceptions, carbon steel typically exhibits greater hardness compared to most stainless steel grades.
I think it is safe to say in the context of carbon barrels and stainless steel products designed for the cleaning of barrels or other carbon steel products the alloy used would be softer than the carbon steel it is designed for.
I am aware having spent over 45 years in the aerospace industry that in special applications fasteners etc. were manufactured in stainless to be extremely hard.
In those cases the price for these items were crazy expensive.