Reminds me of a story where someone had to move a server physically without downtime. The story is on Reddit, but if you really want to read it now there's an archived version. And there's a Hacker News thread too.
That was definitely a thing back then. I've had sites slashdotted several times and never had any issues using FreeBSD and static-ish content and/or compiled C cgi's w/mysql. It also helped that I worked in networking and always had (sometimes just) enough bandwidth available.
Then there was that time in my Atari 800 was slashdotted. The Atari could serve several hits/s on it's 9600 baud interface but the 3com CS-210 terminal server would crash about every 6 hours. https://kl.net/atari/
Probably the most surprising (for it's time) hits/s I had to deal with was when Art Bell had my solar status monitor gif's inlined right on his homepage for a while in 1997-98. It was pretty wild when he would say on the air "Go to my website right now!"