Heat oil in a medium saucepan over medium-high heat. Add leeks and a pinch of salt. Partially cover, and reduce heat to medium. Cook, stirring occasionally, until leeks are soft, 6 to 7 minutes. Stir in beets, carrots, bell pepper, and stock. Raise heat to medium-high; bring to a boil. Reduce heat to medium, and simmer, partially covered, until beets and carrots are tender, 50 to 60 minutes.
Stir in lime juice, and season with salt and pepper. Divide soup among 4 bowls; garnish each with chives and a lime.