It is possible to grow vegetables without soil. Soil provides a method to transfer the minerals and nutrients plants need to the roots of the plant, and also supports the plant nutritionally. However, soil can be replaced with a nutrient-rich water solution--this is the basis of hydroponics, or soil-less gardening. Vegetables can be grown using this method.
Clean out the aquarium with a mixture of 1 tsp. of household bleach to 1 gallon of water. Rinse thoroughly. The water should be shielded from sunlight so algae doesn't grow. Place the aquarium in a box or wooden container and spray-paint it black, leaving a thin vertical stripe on the side so you can see the water level.
Cut a piece of wire mesh that fits across the top of the aquarium. Cut a slit in the shape on an X where the plant will be placed. Fold the edges of the X back a bit so the plant isn't crowded and you aren't poked with the wire.
Place an air bubbler in the aquarium--the kind used for fish works fine.
Fill the aquarium with the hydroponic solution per package directions. Select the type of solution based on the vegetables you're growing and the growth stage of the vegetable (ask about this at your garden center).
Turn on the air bubbler.
Place the plants in the holes with their roots hanging down. If you use nursery plants, rinse off the soil from the roots. Support the stem of the plant with cotton tucked around it so the stem doesn't slide down the X-shaped slit into the water. The cotton should not touch the water.
Place a 1-inch layer of mulch or wood shavings on the wire mesh beneath the plants to support them (you could also use shavings sold as litter for rodents). As the plants grow, increase the layer to 2 inches. Do not get the mulch or wood shavings in the water.
Leave 1 inch of space between the top of the water solution and the bottom of the wire mesh for young plants. As the plants mature, increase the gap to 2 inches.
Change the hydroponic solution once a week for new plants, increasing to twice a week for more mature plants.