like $5-10/month
source: used to have a textboard site that is now defunct
shared hosting is cheap
managed hosting, VPSes, etc... small differences in price points
also make sure you use domainsbyproxy or whoisguard when you register the domain
but text boards barely use any resources, compared to other kinds of sites that have higher-bandwidth/higher-disk usage media (such as images or videos)
however, you will have to worry about moderation and security
so even though the monthly cost is low in dollars, the monthly maintenance and time requirements are high in other areas
you really don't want your site to become a haven for spam or bad content, so you gotta moderate it
also, you can always go fancier/more expensive, like with a CDN or something to protect against DDoSing, since a single shared server won't be able to scale well to handle lots of users, but the fact of the matter is that scalability issues aren't really a concern for tiny sites that barely get any traffic to begin with
ALSO: if you don't develop board software on your own, be prepared to deal with bugs and a lack of support
you might install text board software, and then one day, the maintainer abandons it, and you're on your own
don't use old software, it can get hacked
but if you're a web developer it's no big deal
hell, even put it on github and let the community help you maintain it
also please use TLS, such as with Let's Encrypt or Comodo or something
I hate the lack of HTTPS on 4-ch