re: What's the best way to save streaming video?

It likely has DRM and it may even block screen recorders. You may or may not be able to get around the blocking by turning off hardware acceleration in your browser. Aside from that it's hard to figure out without knowing what kind of stream you're trying to record.

But the fatal flaw with all DRM is the analog hole (anal-hole for short). As a last resort you may have to record the playback with another device.