Just about anything can be grown indoors without natural sunlight. It depends on how serious you are. There are a few different options if you were willing to setup a few lighting fixtures. The cheapest setup (for peppers and lettuce for example) would involve buying a simple fixture and putting a full-spectrum CFL bulb in it. A full spectrum bulb (around 5000K, and roughly 1000 lumens is a good start) closely resembles the sun in terms of specific wavelengths that plants require to photosynthesize.
I myself am currently experimenting with growing basil, green peppers, chives, and spinach indoors in a hydroponic setup with no direct sunlight. I'm at about week 5 and so far everything is doing great; the little bit of basil I have sampled so far tastes great! If you preferred to use a soil setup instead of a soil-less one I imagine you would have similar results.
Just about every fruit has been grow indoors nowadays with the right lighting. Take a look at this impressive little setup growing indoor strawberries!
Note: Growing basil and peppers indoors is a lot different than growing strawberries/other plants solely indoors and require a lot more than just a simple bulb and a spot light fixture. For more difficult plants a more complex setup and a lot of research is required.