Wikipedia's History of Firefighting includes the following snippet:
The first Roman fire brigade of which we have any substantial history was created by Marcus Licinius Crassus. Marcus Licinius Crassus was born into a wealthy Roman family around the year 115 BC, and acquired an enormous fortune through (in the words of Plutarch) "fire and rapine." One of his most lucrative schemes took advantage of the fact that Rome had no fire department. Crassus filled this void by creating his own brigade—500 men strong—which rushed to burning buildings at the first cry of alarm. Upon arriving at the scene, however, the fire fighters did nothing while their employer bargained over the price of their services with the distressed property owner. If Crassus could not negotiate a satisfactory price, his men simply let the structure burn to the ground, after which he offered to purchase it for a fraction of its value.
IMO, this page is being too kind to Crassus (the same one) as it appears to suggest that he was providing a public service for a fee. His own wiki comes closer to the more widely accepted truth which is often recounted in economic history books:
The rest of Crassus's wealth was acquired more conventionally, through traffic in slaves, the working of silver mines, and judicious purchases of land and houses, especially those of proscribed citizens. Most notorious was his acquisition of burning houses: when Crassus received word that a house was on fire, he would arrive and purchase the doomed property along with surrounding buildings for a modest sum, and then employ his army of 500 clients to put the fire out before much damage had been done. Crassus's clients employed the Roman method of firefighting—destroying the burning building to curtail the spread of the flames.
In 6 CE, Augustus, possibly building on Crassus' idea, organised a group of slaves into the Vigiles, a force which functioned both as firefighters as well as policemen of Rome:
Every cohort was equipped with standard firefighting equipment. The sipho or fire engine was pulled by horses and consisted of a large double action pump that was partially submerged in a reservoir of water. The Vigiles designated as aquarii needed to have an accurate knowledge of where water was located, and they also formed bucket brigades to bring water to the fire. Attempts were made to smother the fire by covering it with patchwork quilts (centones) soaked with water. There is even evidence that chemical firefighting methods were used by throwing a vinegar based substance called acetum into fires. In many cases the best way to prevent the spread of flames was to tear down the burning building with hooks and levers. For fires in multiple story buildings, cushions and mattresses were spread out on the ground for people to jump onto from the upper levels.
The Vigiles were commanded by a Praefectus vigilum (Prefect of the Watch) who was appointed by the emperor.
Some sources state that Augustus was inspired to create the Vigiles based on the innovative use of water pumps to combat fires in Egypt.
There do not appear to have been any organised civil emergency services such as paramedics until the 19th century. While flooding was an issue in the Nile valley, I suspect that it was too regular to have necessitated an emergency service. While Japanese earthquakes and tsunamis were meticulously recorded, there again do not appear to have been any dedicated emergency services to provide relief during disasters.