Executive summary: dish soap, bleach solution, a mix thereof, or Star San are excellent options.
Pobrecita has the right idea, but if you want them to last as long as possible (and save money), you will use a better wash than vinegar. Ideal characteristics: it should be nontoxic, it should be cheap, it should kill microbes (both spoilage and food poisoning), it should have a good shelf life, and it should have enough cleansing power to wash off pesticides and waxes.
Convenient solution: a very dilute solution of bleach and dish soap, rinsed afterwards. Bleach is nontoxic in low doses--we even drink it, especially when hikers need to purify stream water. UC Davis writes about how to do this, and they suggest 30 mL bleach (1 tablespoon) per gallon (yielding 200 ppm chlorine), rinsed after a one minute soak. Adding dish soap should not hurt the effectiveness (and may increase it due to increased wetting and dislodging of soil), and adding vinegar (add to the water, never add to the bleach) will strongly increase the effectiveness. This mix has no shelf life and is consumed by use, so discard it afterwards.
If you have fish, you might be able to use a better solution: chlorine dioxide and dish soap. Chlorine dioxide has less smell and taste than bleach, and is similarly nontoxic. But please rinse it to get the dish soap out.
Homebrewers: wash the vegetables with Star San (acid anionic sanitizer) or iodophor. Don't rinse. PAA is also used for this purpose in the food processing industry.
Perfectionists: buy a sanitizing vegetable wash that's targeted at restaurants and vegetable processors. (The stuff in the supermarket is just soap.) This ecolab vegetable rinse--I found it online and this is not a recommendation--meets all the requirements and more, except that it's almost certainly expensive. This choice and brewing sanitizer are the only options that don't need to be rinsed off.
Since Star San (or the upcoming competitor Stellar San) is low cost, kills spoilage and pathogenic microbes, can be stored in a sealed container until needed, it is cheap, and comes in small bottles (diluted to make lots of solution), this is the best option by just about every criterion.
What do I use? Soap. Washing away the microbes is good enough for the amount of time I store my produce.