Can you shoot steel shot in an older Stevens 311 shotgun without damage to the barrels?

No.

yup, hasn't hurt my two in all the years it has been mandatory on waterfowl

Sort of like driving your 1940's Ford at interstate speeds. Will do just fine for a while, but eventually something is going to be damaged. Newer guns use harder steel for the barrels and even they get more wear from steel shot than from the old lead shot or newer (more expensive) soft shot.

You can shoot steel out of the shotgun as long as the barrel has not been backbored. The steel shot remain in the wad cup of the shot until it hit the choke at the end of the barrel, so the shot does not actually touch the sides of the barrel. (this applies to all shotgun barrels).

Make Sure your Chokes are not FULL, but Mod or Improved if your going to shoot steel