After the internet fails, you'd quickly learn how valuable the public library is. They have lots of disc based games for Xbox and Playstation meaning no internet required. Lots of movies on DVD and Bluray, and still pretty good collections of music on CDs.
The smallest public library also probably has more books than you could read in your entire lifetime.
For the majority of disc-based media, any large store would do the job too if we're talking post apocalypse situations but a lot of games these days are absolute crapfests without the plethora of updates. As for servers, I did specifically mean places to download from, not stream. I'd already be downloading every film and TV show I could think of I might like or have already seen and think I'd like to rewatch someday.
That said, the internet is also my means of satisfying my social needs since all of my friends live in different countries. Without that and them, the dire loneliness sets in real quick like.
And libraries have a lot of books, I agree and I already enjoy reading but I have relatively niche tastes that would likely only be catered to by large libraries so the vast majority of a small library's collection would be worthless to me. I know the end of the world is more a "any port in a storm" kind of situation but if I'm just running down the clock before the last human alive becomes the last human to have lived, I'm not really looking to pad that time out by reading How to Win Friends and Influence People or 15 different DIY books.