Season beef with salt and pepper. Heat oil in a 6-qt. pot over high heat. Working in 4 batches, brown beef, about 3 minutes per batch. Be careful not to overcrowd pot so meat will get a good sear. Using a slotted spoon, transfer beef to a plate.
Add onions and garlic; cook until lightly browned, about 5 minutes. Return beef to pot; stir in tomato paste. Cook, stirring frequently and scraping bottom of pot with a wooden spoon, until tomato paste is caramelized, about 12 minutes. Add oregano, chili powder, chile pequín, paprika, Tabasco, and cumin; cook, stirring frequently, for 1 minute.
Add 5 cups water; bring to a boil. Reduce heat to medium-low; simmer, stirring occasionally, until meat is tender, about 2 hours. Stir in masa harina; season with salt. Simmer, stirring, until thickened, about 5 minutes.
FOR SERVING: Chopped onion, cheddar cheese, freshly chopped cilantro, lime wedges, crema or sour cream, tortilla chips