I run it on an AMD FX-6300, with an R7 260X GPU (2GB), 4 GB RAM, Win 7 Pro 64 on a small SSD. Probably doable for 400-500 USD without peripherals.
It's definitely NOT a dedicated gaming PC - it's my home office workhorse - but Elite runs perfectly smooth with the preset high graphic settings in 1920x1080 and still looks good. (Good, not awe-inspiringly great, but really good.)
That's not a recommendation, AMD CPU's are often frowned upon by real gamers, and with good reasons. But if you don't care about high-end graphics, like to play older titles and need a decent, affordable PC for other stuff too, it's something that works quite well.
I do have an illuminated gaming keyboard & mouse combo (bought for typing and graphics, actually, not for gaming), which is an unnecessary personal luxury, and a HOTAS (T-Flight for now, X-52 or -52 Pro or -55 soon) I only bought for Elite. I'd recommend having one, though keyboard and stick (HOWASDAS) works very well, too. Can't speak for gamepads/dual flightsticks/keyboard & mouse.