I get the "Steam servers are down" thing frequently. So immediately when it happened this time, I popped over to Steam, made some changes, looked at my friends list, and fired up Doom Eternal. It all worked fine. So I suggest that Bungie look into the integrity of their connectivity with Steam. Given all the other Destiny-specific connection issues people are having, I wouldn't doubt that there is some kind of infrastructure problem at Bungie's end. If I was Steam I wouldn't be too happy about Bungie pointing the finger at them all the time.
And HaraamPolice, thank you if you do respond. I'd be happy to post my upload and download speeds as well as ping, latency, and packet loss statistics from various web tools I can run almost immediately.
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If it keeps popping up just disable the overlay