What is the cause for hearing the turn signal as soon as you crank up the car without having turned it on this is a 1998 Honda Accord LX?

The sound you are hearing is a relay clicking on and off. If your turn signals work fine then its probably not that one. While you can hear the clicking remove the relays one by one until the sound stops and that will tell you which system is affected. In my case the low and intermittent setting for my wipers did not work and the switch was turned on without my noticing it so the wiper relay clicked constantly without the wipers going. I did not realize that it was the problem until I bumped the switch into the proper off position and the sound stopped.