I looked at Google maps and imagery. First, this square mile thing is misleading. At the far NW corner of the urban area is what you and I would call civilization (bars, restaurants, homes). The other 90-percent of the city limits is wooded areas, cabins, and hunting accommodations. I can only assume they incorporated this rural area in....to pay for city water and police/fire protection.
Second, when you view for pubs/eating establishments....a heck of a lot for a population of 700 to 800 people. If I tried to compare the area to Tenn/Ala....with that many establishments, local population would have to be in the 2,000 range.
The only "eating establishment" or "pub" is a convenience store that sells breakfast muffins.
All of the posts and articles calling Columbia a "city" are nuts.
It is the boonies.
A two lane blacktop highway runs long the west side, near the Connecticut River, which is the border with Vermont.
I've driven most of the roads in Columbia, which is easy as there is one blacktop road, that goes east from NH route 3 for about 3 miles, then goes north to Colebrook to NH route 26. There are a few dirt side roads and that is it.
NH snowmobile trail route 5, which I have also ridden a few times cuts through Columbia. Many of the houses and camps are spaced far enough apart that you can't see one from another.