Surely you can find someone, somewhere, willing to part with an old system, even if it's a PC, to use as a home theater box. A laptop with a broke screen is usually pretty cheap to obtain as well - just hook a monitor up to it to get it going, then toss Windows + iTunes on it. It can run with just power and network connected to it (or use wireless, if the wifi card is N or greater). From there you can use an iPad to remote desktop to it and control it.
A home server box doesn't need to be anything fancy or high powered - it's a glorified file server. I've typically used core2duo (or the AMD equivalent) machines for my home server boxes with success. A laptop with a broken screen can be particularly effective because they're usually cheap to obtain, they're made to handle heat in small spaces, can be placed someplace inconspicuous, and even a USB 2.0 external drive can be used to store & stream the movies. Otherwise, yeah an older Mac Mini is a great option.
But there's not much of a way around the fact that you need something there at your house to do what you're trying to do. Of course, you can just buy all your movies through iTunes. You can also buy movies with digital copies and they'll be available to stream from iTunes, but that doesn't help your existing library, and not all movies offer that option. The ones that do it's hit and miss whether they provide an actual iTunes copy vs an ultraviolet or Fandango copy, and then whether or not it'll be in HD or SD - they used to always be SD, but lately all the ones I buy are HD digital copies, so it just depends. Plus if you buy a movie that's been out a couple of years, the digital option typically has an expiration date so the code isn't even usable anymore.