Jeez guys. Nobody bats an eye around here at the prospect of DVRing a broadcast TV show, a cable TV show, or even premiere content like HBO. It's not like the guy is looking to tap into government feeds and catch Joe Biden secretly say something stupid (as opposed to all the stupid things he says publicly)
To the OP, what particular websites are you looking at? If there is no DRM used, chances are that the files are temporarily stored on the computer while streaming. Perhaps a google search of that program/website will uncover more info.
If it is flash based playback, those are pretty easy to copy and there are a lot of add ons for browsers to do that.