Hartford, Connecticut’s capital city and hub of the state’s insurance industry, is edging closer to joining a small club of American municipalities: those that have sought bankruptcy protection.

The city’s​ ​$49.6 million budget hole and the impending departure of one of its biggest employers, Aetna Inc.,​have shined a light on its unusual predicament: Half of the city’s properties are excluded from paying taxes because they are government entities, hospitals and universities.

It...