A noise gate just mutes the mic when you aren't speaking, mostly to eliminate hum. It won't really do anything to eliminate ambient noise other than when you aren't speaking.
The issue is that OBS takes the raw mic input. It doesn't process it at all like Skype or Teamspeak do. The easiest way to fix ambient noise problems is to remove the ambient noise. The second-easiest is to move the mic closer to your mouth, and turn down the input level. The stronger signal means that the ambient noise will end up weaker, or inaudible.
Failing that, you could try the Mic DSP Plugin:
https://obsproject.com/forum/threads/microphone-dsp-plugin.21034/
Down side, you won't see the mic level activity on the Mic bar, it'll be rolled into System. Mute/unmute still work through the Mic channel though.
Honestly, from the VOD linked, I can't tell which is your voice and which are your friends'. Is yours the extremely acoustic-reflection heavy one?
