You absolutely can run Fediverse software yourself. That's how the internet was created and we have you claw it back from big Tech.
First you should figure out what you want to run. For a few folks and basic microblogging, GoToSocial is super easy to set up, but it's limited.
Bonfire is an amazing fediverse app and it's getting better everyday. They're trying to make it a platform for communities and it's very extensible.
Emissary is also amazing platform for fediverse stuff and then there's HTML and CSS editor to really build what you want from there. Like bonfire, but you have to really create the site you want.
If you want reddit style, I like piefed over Lemmy.
Forums with fediverse? Nodebb.
Pictures only? Pixelfed
Live streams? Owncast
Hosting videos? Peertube
And it all interconnects. So you could run Bonfire, and follow someone on Peertube and comment and like from Bonfire.
It's overwhelming. Id suggest write down what you're wanting and then look at each project and see what fits.
I run my own bonfire at btfree.social and a public peertube at tubefree.org.
Happy to help however I can. I've done some install videos and going to do more soon.