Could be other things, but is usually the fuel pump on something this old.
It could also be a wiring problem to the fuel pump. Broken wire, corroded wire, etc. You need to make sure power is getting to the fuel pump.
There is a relay for the fuel pump. A relay is essentially an electronically activated switch that allows a low amperage circuit to activate a high amperage circuit. Relays can fail.
It could be a computer issue. The computer tells the fuel pump when it should be on.
The pump doesn't run continuously when the engine is not running. After the fuel system builds pressure, the pump should stop until needed.
Are you positive there is no fuel pressure? On the fuel injection fuel rail there is usually a Schrader valve used for maintenance. Using a rag to catch spilled fuel, open the valve to see if there is any fuel pressure. Should be around 50 to 60 PSI, so the fuel should be under pressure.
If you are sure power is making its way to the fuel pump and that there is no fuel pressure, chances are good that you need a new fuel pump.
Not an easy fix for the average shade tree mechanic as you usually must remove the fuel tank to access the fuel pump. However, on rare occasions the vehicle manufacturer will install an access panel for the fuel pump. The fuel tank is located under the car, under the back seat. I would take the time to remove the back seat to see if an access panel exists.