Rinse jicama. Cut off and discard skin and any tough fibers. Cut jicama into matchstick-size sticks 2 to 3 inches long.
Rinse onion slices with cool water and drain.
With a knife, cut off and discard grapefruit peel and membrane. Over a wide serving bowl, cut between inner grapefruit membranes and lift out fruit segments. Put fruit in bowl. Squeeze remaining membrane over bowl to collect juice.
Drain grapefruit segments; save 1/2 cup juice and keep the rest to drink. Combine the 1/2 cup juice with vinegar, basil, garlic, and 1/4 teaspoon salt; return to grapefruit. Add jicama, radishes, and onion and mix.
Peel, pit, and thinly slice avocado onto jicama salad. Gently mix and add salt and pepper to taste.