You could do something like that using point-to-point wireless links or just cables slung between buildings to connect boxes running a self-organizing mesh network protocol like yggdrasil. But there are too many challenges for me to go into depth here ranging from getting buy in from enough people who are located in close proximity, managing user expectations of speed, making services available over such an overlay network (or managing and paying for proxies that provide access to the regular Internet), dealing with geography, etc.
You'd basically be looking at replicating freifunk or nycmesh or doing something along those lines. NYCmesh as I can tell operates more like an ISP so I would expect it to be at least harder than what they do.
Imo time is better invested in developing and advancing decentralized applications and protocols, such as developing stuff using bittorrent/DHT or I2P which can just take advantage of the existing internet.