You can put an html video tag and call it a day, but you'll be missing out on using the best codec for each viewer and bandwidth adaptation and (last I looked, hopefully I'm wrong) usable UI.
Bandwidth is an issue, although I've seen enough high bandwidth, unmetered server offers that I think it might work. Depending on where your viewers are and where you find cheap bandwidth, you might get poor performance just from distance, whereas YouTube and Facebook have CDNs with nodes everywhere.
A seriously crappy PC, poor bandwidth an some noisy old disks is enough to host 5 TB+
So you go for a reseller which there are many stellar ones but they'll either utilize a program with the datacenter akin to an reseller affiliate program with them being the 3rd party support or do it all themselves. If they do the reseller affiliate program they cant really offer anything outside of theie markups on the existing offerings by the datacenter. If they do it all themselves then it becomes much more expensive for the upfront costs.
What you want is a VPS and a CDN which provides a better experience and what every streaming platform uses. Not that expensive either!
Also, is Hetzner actually unmetered or do they claim they are with an asterisk?
Is there a way to do it with some p2p à la popcorntime?
Then if there was a surge, the peers could absorb some of it.
Does peertube do that?
other replies mention the client compatibility, quality and bandwidth issues with self hosting but i think this can be reasonably adressed with just going for a common denominator on the self-hosting side (eg. 720p h264)