I have a 68 Dart 2 door sedan Slant 6 stock with 57,000 original miles.
Having removed the spare cover and trunk mat I can hear a faint sound
from the rear end. It's not a whine or roar but a slight shushing sound
that occilates with the wheel revolutions. There is no sign of rear end
leaking.

Is this just "old car" noise?

Could it be caused by the car sitting and causing flat spots on the bearings?

Should I install a radio that works, put the spare and cover and trunk mat
back and forget about it?

Are you sure it's coming from the rear end? Brake shoes that are slightly dragging sound like that. With automatic adjusters, that's a normal condition.

Thanks,

Drum brakes... It's been such a long time I forgot about that.
I'll take it out to a parking lot and do the "back up and brake hard"
to see if they will auto-adjust on Sunday!

I had bad outer axle bearings in my 75 Valiant. Took me years to figure out what it was, it got louder over the years. Finally I realized it made the same sound on pavemenet, concrete, and gravel roads. Applying the brakes was the only thing that made it quiet up. I finally replaced the axle bearings and I could hear the radio again!

Oh, and the dragging brake shoe could be it too like others mentioned.

Oh, I just remembered a simple way to check for really bad bearings.
Take the back hubcaps off and drive for 3 or 4 miles with little
or no braking. Get out and put your hand (carefully) on the center
of the wheel. If it's hot, the bearing is in bad shape.

This worked on my 64 Newport so I could figure out which side was bad,
of course the bearing race broke (I said really bad) in the next few days.
Now that is a sound that can't be ignored

My bearings were pitted, made noise but did not get hot. If they get hot, you have serious bearing problems.

What causes them to fail?

Ive had mine replaced once already due to failure, and I was assured the other parts were in fine shape, now theyre roaring again. The oil level is OK and Ive only put about 10K on the car. The seals are starting to leak.

_________________
I've been calling it as i see it for my entire life and that's not about to change. Take it or leave it.

I don't know why the failed. Factory bearings that were 19 years old. The balls and races looked rust pitted from water. Those are sealed bearings in the 7-1/4" axle. They do not get oiled from the gear oil. They are packed with grease at time of manufacture.

If memory serves me well, I think if the seals are leaking that is a sign
of too much rear end oil traveling down the axel tubes to the bearings.

The process for replacing a bearing included packing the new bearing with
the appropriate grease, a fun, messy process involving a fistfull of grease
and mashing it into the new bearing...just like a front wheel bearing packing.

If the bearings were replaced without being packed (replacement bearings
have no grease) and the person replacing them thought that the rear end
lube was intended to lube the bearings....well 10 k would be about right
for bearings running on much too thin lubrication.

Just an old mans theory