The Savage/Stevens 311 is a good reliable doublegun. Okiecruffler can give better input but here's a few things to look for....
First, of course, ensure that it's empty.
Look it over from bow to stern for stuff like dings and rust. Shake it a bit and see if you hear anything moving around.Run a finger into the chambers and see if there's roughness.
Check the muzzle and see if the crown is dinged, and if there's any sign of the barrels having been hacksawed.
Hold the thing up to the light and look at the spot where the action and barrels meet at the standing breech. Any light coming through can indicate a headspace prob. If the thing will close with a layer or two of scotch tape on the face of the breech, there IS a headspace prob.Also look for tiny cracks where the standing breech and the water table meet.
Dismount the barrels. Hang them by the lug and tap them with a pencil or other non marring semi hard object. If they ring like a bell, all's well. A dull thud means they're separating. Pass on anything that thuds, too expensive to fix.
Look at the standing breech and see if there's any peening around the firing pin holes. A little isn't that bad, but if it's all on one side it can indicate ignition probs.
Early 311s may have had 2 9/16" chambers.
HTH...