As others have stated, you can absolutely do it. However, your experience running macOS on a Hackintosh will not be nearly as smooth as running macOS on Apple hardware. If you are not technically savvy, or willing to put in the effort to learn, then you're much better off saving up for a proper Mac (have you looked at Apple's refurb site by the way?)
Issues you will encounter when building and running a Hackintosh PC:
* Limited hardware compatibility, especially if you're looking at running it on a laptop (eg: won't support many laptop's built-in WiFi, so you'll either end up having to replace the laptop's WiFi adapter or use a USB adapter)
* macOS updates will often break your Hackintosh, requiring you to apply patches and fixes after each update.
* Problems with Messages, FaceTime and several other apps due to your machine not having an Apple hardware serial number and UUID (there are workarounds for this, but they involve spoofing serial numbers).
* No guarantee it will work with future macOS updates (which is what you've run into with your Mac mini anyhow).
Note: for the Hackintosh fans out there, I'm *not* saying that Hackintoshes are bad per se. I've built a few in the past. But they're not for everyone. If you don't enjoy getting your hands dirty with a build and debugging software/driver issues, then a Hackintosh isn't for you. In my case, I had fun with my builds, but in the end I looked at the time I spent, multiplied that by my usual hourly rate, and realized that I wasn't saving any money over a real Mac. But for the experience/fun of it? Totally worth it.
(Also: you'll be hard-pressed to find a legitimate company who will build you a Hackintosh - they'd be a target for Apple's legal dept. But you may find an individual willing to help you "on the side").
Highly recommend that you have a look at Apple's refurb site and see what you can find there... if you can hold off you will see new Mac Minis showing up there within 3-5 months I'd think.