Your car is a '91 so it doesn't have an immobilizer. You don't have some after market security system that might have failed, do you?
The fact that your car is 24 years old and the dash lights went out makes me think that its probably the ignition switch. A main relay failure would not cause the dash lights to go out. The ignition switch has two circuits in the running position. The ECU and some other stuff are on the IG1 circuit and a bunch of other stuff on the IG2 circuit. The contact in the switch that supplies the IG1 circuit could have failed while the IG2 contact remains operational. It is also possible that the
#29 fuse that supplies the IG1 and starter circuit in the ignition switch has blown. If you can crank the starter motor, then the
#29 fuse is OK and the problem is likely in the switch itself. If the starter motor will not operate, then the #29 fuse may have blown. If the
#29 fuse is blown, then you could try replacing it and see if that solves the problem; however, a blown fuse is a just a symptom usually indicating that you have a short somewhere.
Try wiggling your ignition key and switching from the ACC to the run position. Sometimes that will get the switch working temporarily. If it does, that's great and plan for immediate replacement of the switch.
Its also possible that the switch is OK and that you have a broken wire or bad connection some place. I have seen the wiring under the dash on the NSX. Trust me when I say that a failed switch is the absolute best possible scenario!