So far I know of Github pages and Gitlab pages where you can host a blog for free.
I think medium you can also publish a blog on there, but never actually tried it.
What are you trying to do?
- Static files? Then GitLab/GitHub pages will be fine. There are also many providers like Netlify that specialized in static sites.
- "Live" application like self-hosted WordPress? Then you will need to find a server provider where you can configure and run it.
- Managed hosting of open source software. If you don't want to maintain the blog software yourself it is pretty easy to find people that host common blogging software like https://wordpress.com.
- SaaS blogging. Then you just sign up like https://medium.com, https://blogger.com or https://tumblr.com
why'd this get downvotes
Recommending Medium and Blogger in a privacy community is terrible advice. It's disappointing that this is actually getting upvoted.
Edit: The other two comments provide actual private recommendations. This comment would be fine outside a privacy community.
Because in spite of the question ("private host" as opposed to "privacy friendly host"), people have commented in the context of privacy (probably because of the community we're on); and in that context, the mere mention of github or gitlab is met with downvotes.
IMHO using github or gitlab for that isn't much of a concern privacy wise (at least not more than browsing a website from the companies behind those services), but it surely is when it comes to data integrity (which is a totally separate concern, but people often confuse it with privacy, like they also do with security). I would not trust either service with that.