I am, as I tweeted, so very confused by this:
I wasn’t even sure whether to post here or in GQ, but ultimately I decided the likelihood of anyone being familiar with the actual facts in this remote Minnesota town was low.
So: anybody got idle speculation to share?
I think it’s just the name of the store. Perhaps a reference to prohibition.
Google it, lots of hits.
According to the store’s directory descriptilon:
Corner of Hwy 75 and Hwy 200 (3rd St) Halstad, MN
Prices so low it should be illegal!
I’m not sure if it’s relevant, but I notice the town is right on the North Dakota border. North Dakota was a dry state from its admission to the Union in 1889, while some counties in Minnesota were wet until national prohibition took effect in 1920.
Then again, for all I know the establishment might be run by Mr. and Mrs. Illegal.
Don’t those “lots of hits” seem like they were automatically generated by others running the same Google search? Okay, there’s a phone number in the one you linked: but where is the Facebook page? Why no ads? This really doesn’t fit the culture in that region IMO.
Still, it’s a better explanation than anything I’ve got, so maybe.
I used to live not far from there on the ND side. What do you suppose it “doesn’t fit the culture in that region”? People certainly drink out there and have a sense of humor.
I married into a Norwegian-American brood from this area (which is why we were driving through there on Memorial Day). I am pretty snarky, enjoy pranks and sharp quips and so on. My experience vis-a-vis your last sentence? Yes, people from there do drink alcohol. That’s true. <cough> Yup. There are alcoholic beverages consumed from time to time. <whistling softly> Sure are.
Wait, am I inplying they are humorless and stoic? Umm…I think I had better stop right there and take the Fifth. 
ETA: A liquor store with no advertisement of specials, etc. would be a first IME if you don’t count the ones in some places run by state or local governments with mandatory no-nonsense names like “ABC Package Store”.
I think the name gimmick is probably good for word of mouth advertising. But although I stopped in Hilsboro a million and a half times I don’t think I’ve ever been there. (I lived in Grand Forks for 11 years).
Take two Fifths, 2nd one 25% off. 
Google Street View is from 2012 but the building it shows at that corner doesn’t look anything like the OP’s picture.
I Johnnie Walked right into that one! Well played.
My best guess at this point incorporates some of your points (thank you all), but also retains something of my sense of the culture. I can imagine some wag (perhaps, like me, not native to the region) deciding to open a liquor store there. Upon encountering the byzantine laws and bureaucracy inherent to liquor control in Minnesota, they decided something like “seems like almost everything’s illegal, so I may as well just call it Illegal”.
But that was too clever by half, the whole enterprise landed with a thud just like I’d expect (“well, that’s…different”), and the business was built and subsequently failed spectacularly, all since the 2012 Streetview. Which would explain why there are no banners out front advertising Coors Light 30 packs for $19.99, no Facebook page, no Yelp listing, etc.
A friend just emailed me this link: https://www.livecareer.com/resume-search/r/owner-operator-110129647
So I guess it was someone young, barely off the farm, who dropped out of college to start this business. Which makes me wrong about their not being from the area…but the store “outgrew” its location? Pffft. It already looked too big for that size town, IMO. And of course making a resume to look for another job doesn’t match this claim, so I guess it’s just bluster to avoid acknowledging failure.
Hey, you can by liquor on Sundays for going on a year now. We’re practically an alcotopia.
True! And I guess 3.2% beer is no longer a thing?