Usually you'd find a standard hosting plan has the e-mail side of things setup for you - you just create your mailbox, then either use webmail or configure your mail client to access either a POP3 or IMAP mailbox, depending on how it's set up. To physically install all the components as you required, you'd need to be looking at a dedicated server, which would cost more.
One alternative, depending on your Internet connection at home, would be to setup an old machine which can easily be picked up from somewhere like eBay if no local businesses are clearing out equipment, setting this up at home, and using something like DynDNS to point the mail through to it. That way you still have full control over it, but is sat home meaning there's no extra hosting costs.
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