Today I saw a video showing off the first(?) PS2 web browser. No idea on the details. It's not by Sony and you'd think that Sony would want to control that as much as possible.
I couldn't find pictures of the modem online but it basically looked like a sqaure cigar with a short cable to plug into a USB port and a RJ−?? socket on the opposite end.
No idea if you can download games etc. You can plug in a keyboard to your PS2, probably any USB keyboard will do but if you don't have one a virtual one pops up on screen.