I think you are setting yourself up for real problems here, but in theory it';s reasonably simple:
Use OWIN to Self-Host ASP.NET Web API 2 | Microsoft Docs[
^]
But would I do it? No. Not even slightly. If only because you need a domain to make it a "proper" website (if you don't have one you will have to access it via an IP address instead, so you will need to make sure that your home internet provider has given you a Static IP address or it will change each time your router is reset.
Me? I'd find a cheap (or even free, if you don't mind the hassle you get with 'em) hosting service - many of them will register your domiona for you for free as well - and use their IIS server, database server, and almost certainly get domain related emails as well.