When the ignition is turned to ON, does the Check Engine Light (CEL) come on for 2 seconds and then go off? During this 2 second period, do you hear the fuel pump in the gas tank run? When the CEL goes out, do you hear the main relay "click"?
I would NEVER get rid of a great car, like yours, for the reasons mentioned. The solution is likely simple and you would be costing yourself lots of money.
The solution probably lies with the Main Relay, Igniter (ICM), or Coil. Even if you replaced all of them, it's not going to cost too much, provided you buy them yourself and not from Honda. I would also replace the NGK spark plugs, distributor cap, and rotor. Thess are standard preventive maintenance items for a vehicle with 79,000 miles.
Here is where the Main Relay is located on your 1998 Accord:
http://techauto.te.funpic.org/mainrelaydefine.php
This is not difficult to replace--you can likely do it yourself.