I don’t even remember what I ended up using to yoink a copy of a video that isn’t really even a single file. It was a university lecture, and they ended up using some truly obscure system for storing those videos. After a bit of researching, I found the URL for the folder which contained a thousand messy files. Then, I used some clever stream capture plugin to grab all of those files and write the resulting video into a single easily readable file. I think it more or less played the whole video in the background while saving it, and it took a really long time. As far as I can tell, this method preserved the video quality, and that’s why doing it this way was so complicated.

Technically, you could just screen capture what you’re seeing, and I guess that’s the simple approach you were thinking of. The thing is, you’re going to be making compromises on resolution, bit rate and frame rate. Not really a huge deal for a lecture, but in other contexts these things might matter. Also, screen capturing tends to take a lot of space, so you might need to tweak the settings a bit to make sure it won’t take your entire disk.

As others have already suggested, Audacity can do the same thing for audio. Same problems still apply, but it should work no matter where the audio is coming from.