c3shooter is correct in identifying the OP's Bay State shotgun as being manufactured by Harrington and Richardson. They made Bay State labeled shotguns from 1908 to 1942.
There were other Bay State single barrels. Bay State Arms Company made single shot shotguns and marked them "Bay State Arms Company" in the period 1870-1874.
The company reorganized in 1880 and produced single shots marked "Bay State Gun" from 1880 to 1909 when they were pruchased by Hopkins and Allen Arms Company who continued to produce the "Bay State Gun" until 1915.
Joseph Vorisek in his "Shotgun Markings 1865-1940, A List" also indicates that J. Stevens Arms Company also made "Bay State Gun" marked single barrels after WWI.
The Harrington and Richardson and J. Stevens Arms Company manufactured Bay State shotguns are safe with modern ammunition. Those manufactured by Bay State Arms Company are not safe with modern ammunition. Hopkins and Alleln Arms company guns should be safe, but I have no references to positively indicate that the barrels are flluid steel.