According to this list, it seems that most PS2 games that have online play have already been shut down, and I expect that sooner or later, the rest will go away, too.
But I like my PS2, and I have no problem with using an old console, if it's still fun, or useful!
Can I use the network capability for anything else? (I know a few games support LAN play.) Like, is there a web browser, or a way to stream internet radio or Hulu or Youtube, or something like that?