99 Chevy Malibu

You must have disconnected the battery. GM anti-theft systems are very effective at keeping owners out of their cars.

Crank the engine for just a second, then leave the ignition switch in the "run" position. You should see the Security light blinking. When it stops blinking after 10 or 20 minutes, turn the ignition switch off then back to run. Wait another 10 or 20 minutes until the light stops flashing. Do that a third time. When the light stops flashing, start the engine without turning the ignition switch off first. You might want to connect a battery charger on the lowest setting while you're waiting.

If the battery wasn't disconnected, you might look at the oil pressure sending unit. I'm getting out of my element now, but there are some GM models that only turn on the pump for a couple of seconds by the Engine Computer. After that it gets its power from a tap on the sending unit. That turns the pump off after a crash that ruptures a fuel line, but it will also fail to power the pump when oil pressure is low and when it's low due to low level.

Perhaps you can pop the cover off the fuel pump relay and reinstall it that way so you can watch it to see if it's staying activated.

Check to be sure the fuel pump is running. If it isn't, the engine will still run if you squirt starting fluid in through a vacuum hose.

Every manufacturer has a few engines they don't deserve to be proud of, but I'm not an expert on GM's best and worst.

You have a good clue in your observation that the engine runs for a few seconds. That is typical of lack of fuel whether it's an anti-theft system getting involved, a fuel pump that is stopping, or something like that. Some other clues you might look for is how it stalls. The anti-theft system will cut off the injectors instantly, just like when you turn off the ignition switch. A fuel pump that quits will result in a gradual decrease in engine speed and some sputtering over a period of a couple of seconds. There can be a failure of an electrical component in the ignition system, but the engine will run longer and for varying lengths of time. A component that fails intermittently due to heat will not fail at consistent lengths of time.

I'm guessing you tried this already but it bears not overlooking; have you tried holding the gas pedal down 1/4" to see if the engine will stay running? This is a real common problem on Chrysler products after the battery was disconnected, and it has an unbelievably simple fix but you'll never figure it out, even after it seems to correct itself.