Is is possible to see old Instagram stories?

You can only watch your own old stories on Instagram. You cannot watch other user's Instagram story after posting 24 hours later. Story get disappear after 24 hours. 

You can watch your old stories  in archive list. Open your app and tap on the profile tab. Tap the archive tab in the left corner at the top. Here, you can see your old stories. You can save and delete the stories from here.