Selfhosted

It does not need to interact with other Mastodon instances, but users on the local server should still be able to follow accounts on other Mastodon servers.

This sentence is in conflict with itself, follows are "interaction"

This is not possible. The other servers need to be able to communicate with yours, and for that, yours needs a domain name.

If you only care about local users following other local users, and only other local users, then this is possible. But if you add remote users into the mix, you need to federate, and to federate requires a domain name.

Adding to why it has to be reachable: when a local user follows someone on another server, that server pushes new posts to your server's inbox, so the remote side has to be able to resolve your domain and connect over HTTPS. A LAN-only box can't receive that.

The least painful compromise is a cheap domain (or a subdomain you control) plus a tunnel or a small VPS acting as reverse proxy, so nothing on your home network is exposed directly. Then close signups so only your household can register. If Mastodon feels heavy for a handful of users, GoToSocial is much lighter on RAM for that use case, though you'd use a separate client app with it.

One thing to decide up front: the domain is baked into every account and can't be changed later without starting over.

~~littlefedi is out as of coupla days ago. still in early stage but should cover your use case. local binary, world comms via "lighthouses" don't need domain, vps, etc.~~ there's something strange afoot wrt the author's willingness to be forthcoming about the project's provenance. I'd stay clear for the time being.

You can use a subdomain, and I'm pretty sure you can get a subdomain for free. It won't be pretty but it will do what you want.

ActivityPub doesn't work that way, but you can follow public Mastodon accounts using RSS.

Look into snac2 or littlefedi as alternatives for Mastodon if you want to save resources. Littlefedi has a specific use case to run off the grid

Have a look at GoToSocial.

I don't use Mastodon, but I don't know if it's possible to do this particular combination:

without owning a domain

users on the local server should still be able to follow accounts on other Mastodon servers

Like, a client that supports having accounts on multiple Mastodon servers would work. Then they have a local account, and they have an unrelated account elsewhere on another Mastodon server and follow people using that, so that the "following" has nothing to do with your local server.

But if you want federated behavior, where a user connects to your server and has their account there tell the local Mastodon server to pull down posts from another users somewhere out on another Mastodon server, you might need to have a certificate on your server that other servers recognize as valid, which in practice would mean one signed by a public CA.

I don't know for certain whether this is actually necessary, but I could believe that it is.

entirely within LAN

Wat? LOL