Hi EmForce,
A DUI by itself does not mean you cannot be granted access at a nuclear plant. There are several other factors that can play a part like did you complete what the court required as the punishment? (pay fines, community service, etc.) Other factors such as a pattern of bad decisions (criminal record, etc.), bad personnel references, bad employment references, etc. play a part in your background investigation and can prevent you from being granted access if not favorable. It comes down to one word.."trustworthyness". You need to prove it. Every utility is different but they all have standards to meet.
I hope this helps!