I don't know what lengths you're willing to go to nor what equipment you have vs are willing to buy, but when you come upon the situation where it won't start, you need to rule out each subsystem at a time - fuel, air, spark, and timing.
We're talking fuel now, so since you know where the fuel relay is, when it won't start you can have one person turning the key from off to accessory or to be cranking the engine while another is measuring whether there is power input at the fuel relay. A test light would probably do this but I think in terms of using a multimeter.
If there is not, you have a problem upstream of the relay. If there is, you can measure for power at the output of the relay. If there is not, you have either a faulty relay or a fault in the circuit energizing the relay.
If there is power output, it's going to the pump. Follow the wiring, backtrace from the pump if that's easier. There should be some place you can access it (pump power wires) without having to pull the tank off yet. A multimeter with insulation piercing probe tips can check for power on the leads to the pump. If there is power on them, and you are not getting fuel pressure at the fuel rail in the engine bay, then either you have a pump fault or a fuel line clog, which would usually happen at the fuel filter.
Another thing some people might do is have one person with their ear down next to the tank to listen for whether they can hear the pump spinning (attempting to pump) when the key is first turned to accessory, or when trying to start the engine but then you have starter/engine cranking noise competing with hearing fuel pump noise.
What I am "guessing" you are describing about a black box in the back is probably an emissions control box with a valve, maybe pressure sensor, and then evap canister or it's just inline to that, none of which should prevent the vehicle from starting and running no matter what condition it's in.
