If you need to stream to many computers at once, Icecast is pretty useful. FFMPEG can directly stream to an Icecast server (alternatively, you can use similar settings in OBS). I capture the x11 window, then encode using VAAPI for reduced CPU usage.

ffmpeg -vaapi_device /dev/dri/renderD128 -video_size 1920x1080 -framerate 30 -f x11grab -i :0.0+0,0  -vf 'hwupload,scale_vaapi=w=1920:h=1080:format=nv12' -c:v h264_vaapi -f mpegts -content_type 'video/mp2t' "icecast://source:password@example.com:8000/name.ts"

• The coordinates after the -i is width, height, left offset, top offset.
• The video filters are to scale to 1080p, while maintaining the vaapi encoder.
• I use mpegts to stream the video, though other file formats that allow streaming such as matroska also work. If you're streaming theora/vorbis, use ogg, for audio, use opus if you're encoding opus, so on.
• I set the content type to video/mp2t, which makes the video stream properly detect. It also makes the m3u file download rather then play in the browser.