Most stainless barrels are 416 or 416R, which is magnetic. 300-series stainlesses are not magnetic, but no one would make a barrel out of them.
Put a drop of 50:50 nitric acid:water in an inconspicuous spot. No rxn = stainless. Fizz and dark spot = carbon or low-alloy steel. This is from an old ASTM STP - Identification of Metals and Alloys by Spot Test or some similar title. Dark spot easily removed with 600 grit paper / steel wool.
Regards,
Andrew