Place plastic wrap on top of a sushi rolling mat about an inch extra on both ends of the mat. Set aside.

Bring water and rice to a boil in a saucepan over high heat. Reduce heat to medium-low, cover, and simmer until the rice is tender and the liquid has been absorbed, about 20 to 25 minutes. Transfer the rice to a large bowl and add rice vinegar. Stir using a wooden spoon and season with sugar. Set the rice aside and let cool for about 30 minutes.

Once the rice has been cooled, place a sheet of seaweed on the sushi mat. Spread a thin layer of rice on top of the the seaweed. In no particular order, place 2 to 3 avocado slices lined across the center, then add the cucumber slices and then the crab meat.

Using the mat as a guide, carefully roll the California roll into a tight log. Cut roll into 6 even pieces using a sharp wet knife. Repeat with remaining sheets of seaweed and filling.