How can you determine if an air compressor tank is safe?
Take it to a shop that either does hydrostatic testing, or has an arrangement with a place that does. It will cost far less than $100, barring a major ripoff. Any place that deals in compressed gasses (welding supply or diving supply) needs to do this on a regular basis (5 years per tank), as do fire extinguisher service shops (which might actually be the place your dive shop sends a tank to for testing.)

The formal tests (in the USA) are based on Department Of Transportation regulations for portable tanks - fixed tanks are not required to be tested (at least by the DOT), but can be. Twice working pressure may be more than is correct - I'm more familiar with dive tanks, and there it may be as little as 5/4 rated pressure - failure is not "burst only"; a tank can fail the test because it permanently deforms (the tank is submerged in a tank of water while being tested - the expansion of the tank when it is filled causes a measured about of water to leave that tank. If the tank being tested does not shrink adequately after pressure is released, the tank fails testing.)

Most fixed tanks fail from internal rust due to lack of proper maintenance (draining the tank regularly.) If your tank is 40-50 years old, it might be worth simply replacing it with a new one, particularly if you have difficulty getting it tested at a reasonable price.