It’s on a VPS. Whether that’s really self-hosted may depend on how much of a purist you are, but it’s fully self-managed, not SAAS.

It’s recommended to have a PTR record mapping your IP address to your domain, which you wouldn’t be able to do with a residential connection from a typical ISP. I do send mail from multiple domains though and I haven’t had issues with deliverability. What I do not send is any kind of high-volume mail, which would likely attract a different kind of scrutiny.