The Swiss guards are generally based in Vatican City.
There is no city within Vatican city. It is a location unto it's self. It is made up of St. Peter's, the Vatican musuems, the Sistine Chapel and the offices/living quarters of the Pope. It includes private gardens, a security force, and palace guards.
The Vatican uses the Euro.
Vatican city
The only remnants of the military of the Vatican are the Swiss Guards who act as body guards for the pope. In earlier years when the Vatican was part of the much larger Papal States, the vatican did have a sizable army. It is highly unlikely anyone would invade the Vatican these days and, if they did, there are many nations that would rise up to protect it.
The Gendarmerie Corps of Vatican City State (Italian: Corpo della Gendarmeria dello Stato della Città del Vaticano) is the gendarmerie, or police and security force, of Vatican City. The Swiss Guard is the personal body guard of the Pope.
The Swiss Guards are housed in Vatican City where they receive room and board as part of their salary.
The worldwide headquarters of the Catholic Church is the Vatican in Rome and is headed by the Pope. The 'headquarters' are in the Vatican city. This is inside Rome but it has its own government so it is not actually part of Rome.
While the pope is the head of state of the Vatican, he leaves the day to day management of the city to an appointed committee with a president and vice president. The committee acts much like a city manager would in most any other city.
The Vatican Security Forces (Vatican Police) are called, in Italian, Corpo della Gendarmeria dello Stato della Città del Vaticano. They are not to be confused with the Swiss Guard who act as body guards for the Pope.
The Vatican is an independent country, a city-state. The smallest country on earth.
In the past the Vatican did have an army. In 1970 Pope Paul VI abolished all the military bodies at his service except the Swiss Guards, the Gendarmerie was transformed into a Central Security Office, with the duties of protecting the Pope, defending Vatican City, and providing police and security services within its territory. Its name was changed in 1991 to Security Corps of Vatican City State and in 2002 to Gendarmerie Corps of Vatican City State.