Nextcloud is probably among your better packaged options. Free Software / OSS:

Share and collaborate on documents, send and receive email, manage your calendar and have video chats without data leaks.

https://nextcloud.com/

For a Twitter equivalent, Mastodon.

Facebook's a bit more of a challenge --- longer-form content, the ability to group contacts, and the like.

Principally there are Diaspora*, Friendica, and Hubzilla, all based on or near the ActivityPub standard.

https://joinmastodon.org/

https://friendi.ca/

https://project.hubzilla.org/

There are limits and compromises to these, though all should handle small-scale (family/community) sharing easily. Mastodon's at millions of users, the others are probably smaller.

At scale, moderation and administration of Diaspora* is a bit of a pain, though that's (slowly) improving.

As others have said Mastodon might be what you want. You can self-host it, either "at home" or "in the cloud", and you can choose not to federate at all, or choose to federate only with instances you choose. You can choose to run as "invite only", or to allow anyone, and then eject people who behave poorly.

So you can set it up as an "invite only social network where I choose who can join", as you say.

Invite-only and self-hosted are different things.

Discord is invite-only but not self-hosted. Maybe that would work for you?

I want both. Basically I want something federated but not connected to the "fediverse" as in Mastedon. Perhaps I need to look into Mastedon.

alternatives:

scuttlebutt https://scuttlebutt.nz/

twtxt https://indieweb.org/twtxt

iris.to (hypercore protocol)

meshtastic https://meshtastic.org/

bbs https://www.ipingthereforeiam.com/bbs/?step=name&all=0

Maybe flarum