looking for some help with an old Rheem 1005 HVAC system. I’ve scoured the Internet for days, so this is a last ditch effort at self-help. I’m installing an ecobee to an existing 4-wire system that had a 5th C-wire already available at the thermostat. The thermostat is hooked up completely but I don’t know if the c-wire is connected at the other end in my HVAC control board because if I turn the ecobee on, it just cycles over and over and Shorts out or something, then immediately kicks back on. The inside of this unit looks like nothing I’ve ever seen before.
I just need to know if that loose blue wire is my C-wire and if so, where can I connect it, or if I need to wire a new c-wire. Pictures are attached for reference. 
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Best Answer
No, the loose blue wire is a 208V tap. The C wire comes off the other side of the transformer. I highlighted the blue 208V wire you referenced, plus the C wire (yellow) in your schematic:
It's one of the two yellow wires, which go to another area (not pictured)
It actually doesn't make a difference which yellow wire is which (AC voltage, so no polarity) -- one goes to the "R" terminal (power, usually red) and the other is "C". From the schematic, you might have only 4 terminals for the thermostat, but there at least does appear to be a hookup for the C wire.