It is pretty straightforward. As I recall from the show, this town was the location of the only viable gas stop between Atlantic City, NJ and New York, NY, and a criminal gang rolled in, outnumbering and out gunning the town's 3 or 4 man police force, and posted armed guards at the roads into town, effectively controlling illegal liquor shipments between the afore-mentioned cities.
It was a cool few episodes, but I can't imagine something like this would have lasted any longer than it took the state police/national guard to gear up and force the criminals to step off, but maybe I'm wrong?
I have commented before about Prohibition's ineffectiveness based off my reading of Last Call by Daniel Okrent. The head of the bureau that was supposed to enforce the prohibition of alcohol reported that they needed far much more money than was being given to them, and as a result, they were unable to effectively enforce it. (In part, this was due to the Depression forcing budgets to be slashed.)
Keep in mind that criminal gangs would pay off the police and the judicial system during this time, so this gang could conceivably be doing that. (I have not seen the episode of this show, so I am just going by your description here. If the plot of the show explicitly denies this, then I find it a bit harder to believe.)
On top of that, crime during this period of Prohibition had skyrocketed and organized crime did take over in a lot of ways. Big cities like Chicago certainly felt it, so a smaller town being overwhelmed is believable.
All of these things combined together make this story fairly believable. (Though as I said, I have not seen this episode of this show and I have forgotten what city the show was hypothetically based on, so I cannot find out if this is based off of a true story.) There are multiple contributing factors as outlined above -- lack of police resources, corruption of police, and the increase of crime across the board.
Edit: If you have a problem with my answer, please let me know why. This post has been steadily downvoted and while I don't care about the vote total, I would like to address whatever you may think is wrong with this post.