In terms of minimum spec rig, I'd probably aim for something like this:

PCPartPicker part list / Price breakdown by merchant

CPU: AMD FX-6300 3.5GHz 6-Core Processor ($105.98 @ Newegg)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($19.99 @ Newegg)
Motherboard: Gigabyte GA-970A-UD3P ATX AM3+ Motherboard ($61.98 @ Newegg)
Memory: Kingston HyperX Fury Black 8GB (1 x 8GB) DDR3-1600 Memory ($27.88 @ OutletPC)
Memory: Kingston HyperX Fury Black 8GB (1 x 8GB) DDR3-1600 Memory ($27.88 @ OutletPC)
Storage: Kingston HyperX Fury 120GB 2.5" Solid State Drive ($42.99 @ Adorama)
Storage: Hitachi Deskstar 1TB 3.5" 7200RPM Internal Hard Drive ($44.00 @ Amazon)
Video Card: Sapphire Radeon RX 480 4GB Video Card ($199.00)
Case: NZXT Source 210 (Black) ATX Mid Tower Case ($36.99 @ SuperBiiz)
Power Supply: EVGA SuperNOVA G2 550W 80+ Gold Certified Fully-Modular ATX Power Supply ($70.98 @ Newegg)
Total: $626.67
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-07-05 17:40 EDT-0400

You could come in under $600 if you've got a case, or hard drive laying around to use. You're also not going to be using the best settings on your five box team like this, but I think for people looking to box on a budget, this is probably a foregone conclusion. You can also get a little extra bang by shopping used. For regular 1080p gaming this should also work just fine actually.

I haven't seen Logical Increments before, but it's a nice little reference. Though, it points out one problem I have with similar listing sites, it's all using the price from Amazon. Because affiliate links.

Which means, you're not getting the best deals on your parts, but the guy who's getting click revenue probably is. I was really hesitant to post on this, because we don't really have a budget here, just a minimum for boxing question. And pinning that down is almost impossible without more information. Like the size of the monitor you'll be using, what graphical settings you're hoping to use, addons, etc. All those things can contribute in a big way to making the whole thing work/not work.

If you are willing to build on a slightly looser budget over time, I would suggest something a bit beefier in general. Especially in the processor department. But, that's the real question. What are you willing to compromise on? Do you want it fast? Do you want it cheap? Do you want it powerful? Can't really have all three, depending on where you're willing to compromise the holy trinity is where a build like this will land.

Also, I'd recomend scouting Newegg and other sites daily for those daily deals/flash deals. You can pick up some good stuff cheap. I grabbed 2 sticks of ram free with my SSDs that way. Saved me a good chunk of cash.