Radio and radar detector wont shut off

Do your dash lights, etc., stay on as well?  There's a short somewhere that's backfeeding power into the accessory circuit.  Have you by any chance done the retained accessory power (RAP) mod: https://www.the12volt.com/relays/page5.asp#drto?  If so, you could have a stuck relay or a problem with the door trigger.  Otherwise, you'll probably have to start unplugging/disconnecting stuff to isolate the problem.  For example: if you unplugged the radio, and the radar detector shut off, that tells you that the radio probably has an internal short.

I'm guessing your capacitor has a digital display?  Is there an accessory hookup required for the display to light up?  If so, disconnect that and see if your problem goes away.  I'm not sure about the beeping (low voltage maybe?- check your manual).