My 2001 XC70 won't start - I think it's related to the fuel system?

Yesterday, I drove maybe 5 miles across town, parked it for about an hour. When I started it up to drive home, it ran very rough and kept dying. The red triangle lit up on the dash with an ominous "check engine" message. The car was low on gas, but the low fuel light was not on. After a few attempts, it finally stayed running and I drove it to the gas station and filled it up. It was running fine after that, and the "check engine" light and message went away.

Then, last night, I drive it to the grocery store less than a mile away. 20 minutes later, I try to start the car and it wouldn't turn over. However, this time, there's no red triangle lit up or "check engine" message or rough running. I walked home, returned an hour later, cranked it plenty but it still wouldn't turn over, not even sputter.

I'm wondering if these two events are related? I'm wondering if it's something with the fuel system, though wouldn't I have noticed rough running or loss of power? Maybe the fuel pump died?