I have a 1985 245DL with 145k miles. Changed wheel bearing grease two years ago, drove about 10k miles since changing wheel bearing grease.

Over the past 2 weeks, I've noticed a shaky feeling when I hit the brakes. Today, I heard a noise that sounded like a flat tire on my right front wheel. The tire was fine. I noticed that the wheel bearing cap was missing (though hub cap present). White smoke was going above the caliper. The wheel bearing grease was brown and oily. No leaks were present anywhere. The brake pads and calipers were ok. The bearing had no scratches on the races, but the bearings did not move very easily.

What is the nature of this problem?

Sounds like the grease was contaminated and stopped doing its job. Get a new cap,new bearings and a new seal (mandatory!). Replace the old bearings, pack the new ones with fresh grease, and adjust the bearing per a Haynes or Bentley manual.