You can run a forum for free using a host like Jcink Forum Hosting or IcyBoards. Both are free hosts that offer ways to get your database.
(Jcink Forum Hosting allows you to download your database at any time for free. IcyBoards costs 10$ to get premium for a month then you can download your database as many times as you'd like within that month.)
Each have their benefits...
JFH's database back up is free but you have to know how to convert databases from IPB1.3 to modern if you want to self-host. (It is a little more difficult than modern hosts...)
IB's databases are locked behind a paywall, but if you do pay then you get a MyBB 1.6.14 database that you can easily slide into any self-hosted MyBB 1.6.14 forum or you can use a converter to go to another software.
Of course if you want to spend money you can get and use a domain on either JFH or IB. But you aren't required to.
Like I just mentioned
here, I do not recommend using free web hosts to run a forum. It will not work out that well. (Expect slow loading times, random downtimes and resource limit hit errors.)
Personally, I would not suggest buying a domain, hosting and a software license for Xenforo/IPB 3.x for a new venture. (Especially not if you've never run a forum before and have no idea how much work is involved. To be rather blunt, getting staff is hard and you aren't guaranteed to have any help for a long time. So you may have to be solo admin for years.)
I suggest getting a free hosted forum and building a userbase before you look at self-hosting. Know that you have an audience before starting to sink money into a forum. Because if your project flops and you've put a ton of money into it... it feels worse. Plus it takes some stress off of running a forum if you don't have to deal with the financial side.
Not everyone has tons of money and when running a forum you can be hit with unexpected costs... Your host closes so you have to scramble for new hosting, even though you may have been paid up for another 4 months.
I would also suggest abandoning the idea of putting your own ads up on your site. Free forum hosts tend not to allow it unless you pay for ad removal (and that can be anywhere from $5-$10 a month, so more than you might pay for hosting). And even if you were a new, self-hosted, forum... having ads up early on is not a good sign. People may avoid your site if they feel like there are too many ads (provided they don't run adblocker anyway).
And even then the amount of money you'd make from having ads will be pretty small. (I mean until you get a fairly large forum, say 50,000+ posts and 1000+ members, you won't make very much at all.)