I lived in LA for 5 years in the late '70's. Some of my favorite places were:
Pinks - for the best chili dogs in town
Tommys - the best chili cheeseburgers (on Beverly & Rampart, north of downtown)
Cassels - great grilled burgers, made fresh'n fast while you wait (get 'em medium rare)
Pantry - amazing breakfasts and sourdough toast, downtown (especially at 3:00AM)
Langer's - great kosher-style sandwiches, NY Style (near McArthur Park)
Bergins - oldest Irish bar in LA (look for my name on a shamrock on the ceiling)
Of course, that was while I was a student on a budget, but for cheap, good food or drink, those places were great! And they are still there!