Good luck with that one.
You can look into standing up an nginx-rtmp or RED5 server on a VPS in a datacenter, but the cost of bandwidth is probably going to hit fairly hard. You'll also need to serve a player (jwplayer is pretty common). At that point, it's all in your hands. It's an expensive and tech-savvy-required option, and lacks a ton of the niceties of established livestreaming sites.
There are a ton of streaming services out there though; Twitch is the dominant name in the market at the moment, with YouTube, Facebook, and Mixer vying for a distant second. Plenty of smaller kibble that generally aren't worth mentioning, and a number aimed more at specific niches (like art-creation streamers). I'd advise against any paid services at the moment; there are plenty out there looking to flat-out scam folks.
Generally though, it's best to stick with one of the biggest four, and just work within the limitations of the platforms, guiding people in that direction over social media. If you don't care about discoverability, YouTube has a lot to offer with higher bitrate caps and quick transcoding, even if they tend to mangle the video.