What the heck is wrong with my TV? (details inside)

I’m guessing some of the control circuitry gave out.

Make sure your TV is set to the right input channel and input source. Since you get the MENU, and blue screen, it indicates that all other components are working on the set. I doubt if it’s a defective Main Board.

“Big box” is CRT type ?
The fact that it plays short bursts of stuff ok then cuts out makes me lean the direction of @XOIIO‘s answer, some kind of control circuit snafu.

Does sound stay on while the picture cuts out or all stop?
If the sound stays on then it could be the CRT circuitry, like the High-V , but you probably would have mentioned a symptom like that.

DO NOT LEAVE POWER CONNECTED IF YOU CONSIDER THE FOLLOWING:
If it’s that old you might be able to open it up and look for further symptoms, reseat connectors and components, and generally gather more information for identifying the cause of the problem. Leave it unplugged for a real minute or so before getting inside to let high voltage drain out of the CRT circuitry.

I don’t think it is anything you as a consumer can fix ( you tried the external, not inside the box fixes.) The TV could be 8 years old or older. The cost to fix and value of the TV could be the same, in other words the time has come to replace it. There are still some near “Black Friday” prices available.

Thanks to all who have answered so far.
The tv is most certainly older than 8 years and it is a CRT type
I tried as @Smitha suggested verifying that it was on the correct input channel and it was.
@dabbler – No, the sound comes on then goes off again within a very brief timeframe and I NEVER achieve a picture.
Since the dvd goes directly into the tv without going through a cable box, should it not act independently, that is, is it not an independent input? And, if so, at what point in the output of the DVD and that of the cable merge? Perhaps merge is not the correct term. They both have to produce a pic on the screen and provide sound for the pic so it would seem that the problem is located between these input points and the final output.

Your further description suggests some portion of the internal power supply in the TV is failing and that’s shutting down the whole system. It could be as simple as some capacitor shorting but could be a more fancy part like the flyback transformer that makes high voltage (thousands of volts) for the picture tube’s electron gun.
Troubleshooting and fixing the power supply might not be that difficult for someone with the background and tools and spare parts lying around. Most of us mere mortals don’t have all that around.
I’d still open the case up and try reseating the parts and cables that are removable, in particular around the portion where the power cord plugs in. Mildly corroded connections can cause all sorts of problems. ONCE AGAIN UNPLUG THE TV FOR A FEW MINUTES BEFORE OPENING IT UP if you are going to try that.