Place beets into a large saucepan and pour in enough water to cover the beets by 1 inch. Bring to a boil, reduce heat to medium-low, cover, and simmer until the beets are easily pierced with a fork, about 15 minutes. Drain and let cool.
Peel beets, cut into 1/4-inch thick slices, and cut slices in half. Place into a bowl.
Smash garlic with salt in a small bowl using a spoon until the mixture becomes a paste. Stir lemon juice into garlic paste. Retain about 1 tablespoon lemon dressing in a small bowl and pour the remaining lemon dressing over beets; toss to coat.
Divide arugula between 4 plates; top each plate with beets, feta cheese, and olives; drizzle salads with olive oil and retained lemon dressing.