Can you start your own fire department?

Theoretically - yes. However - your 'powers of arrest' are limited to those of any ordinary citizen.

Another View: Actually -- no, you cannot. It is realized that the Word 'police' is commonly misused and misunderstood by the general public to refer to any uniformed force carrying out enforcement-type functions. However, the fact is that only the government retains "police" powers and you cannot assume "police"powers on yourself. You can certainly choose to start a business supplying guard services or uniformed security forces but you would not be establishing your own "police" force.