Do Playstation 2's still work while offline using only games disks?

Solution 1:

The Playstation 2 with networking capabilities was shown off at E3 2001, with a collection of peripherals that would enable free multiplayer over a broadband connection (in the US, other regions were provided different adapters). However, the console still had plenty of titles that could be enjoyed fully in a single player or local co-op fashion.

I cannot find information on online only titles on the PS2, because that would answer your question properly. However, there is a list of games on Wikipedia that includes all online multiplayer titles for the PS2. The list is far too long to convert to an answer here, however the list is not overly important to this answer.

To fully answer your question, yes, your PS2 will still work with game disks only, long after the collapse of society and our global Internet infrastructure, just as long as the laser reader holds up.