A website and a forum are complete different things.
You could create a website with
a static web site generator and publish it on your shitty router, because it doesn't take resources at all and it is online 24/7. Or you could ask a friend if you could publish it on their servers, because everyone who manages a server (or v-server) wouldn't even recognize a static web site on their server, because it just doesn't need resources.
A dynamic website like a forum needs much more resources. A Raspberry Pi is the first thing, what comes into my head, but even the newest one could be overpowered and you could even put a mumble server and a game server on it and it still work perfectly.
If you want to do it a little bit more professionally and you expect a lot of people to use your site and want to be reliable (no down time), I recommend to rent a v-server for 10 to 20 bugs. But because of their shared resources, even they are overpowered for a forum alone.
But because your question was "What does it take to run a forum these days?", I want to mention the social aspects, especially the behavior of the people in the 20s, which I listed in
this post, which could get problematic very fast and should be handled asap, because I know people who gave up managing a forum, because of the people.