Live streaming to a disk
If you’re up for building your own server, and have some coding skill, then you could look at setting up a nginx server running the RTMP extension that can capture an re-stream live or on demand video. It can also do near realtime transcoding or "at completion of capture" transcoding using ffmpeg.

For the system I setup, I send live video in using an IP based camera system from multiple sources that can do RTMP streaming, or if I want to capture desktop content like a video game I use OBS to do the streaming to the server.

The actually user website itself runs from a node.js based system that has access to the content recorded by the nginx rtmp capture.

OR

You could look at a pre-build solution like Kaltura which has both a open-source community version and a commercial version. Think of it as your own YouTube server setup that has a tone of features ready to go out-of-the-box.