There's a point in every person's life when they give up on trying to be the cool kid and surrender to their fate of scoffing, "kids, nowadays!" at every opportunity. I reached mine a couple of years ago when Snapchat became a thing, Twitter started being about politics and news more than fun conversations, and everyone (and their mom, dad, kids, and grandparents) was on Facebook. Now I just sigh and roll my eyes each time I see people putting more of the silliest parts of their lives online, which, as I type it, sounds quite hypocritical given this very paragraph. I guess I'm more selective of what I share and who reads it - I like you folks.
Curmudgeon musings aside, Instagram has now released a couple of new features, which I'm pretty sure have been there in Snapchat for a while (Memories, if I recall correctly, I might have even written about it... oh look I DID!), that let you take those ephemeral stories and keep them forever. First is Stories Archive, which has a history of all your past Stories so you can browse and relive those fascinating moments again. These are only visible to you and are sorted by chronological order (newest at the bottom). If you don't like it, you can turn off auto-archiving.
Stories Archives. (Excuse the iOS screenshot. As always, Instagram only provides these.)
The second feature is Stories Highlights which lets you pick anything you've shared to previous stories and showcase it in grouped highlights on your bio page. You can have as many highlights as you want and give them individual names. Oh and these play out like a story when people tap on them.
Instagram says these have rolled out to the Android app with version 25 on the Play Store, but I can't confirm they're there because I have 0 past stories to highlight or archive. Let us know if they're live for you.