In my town what you're proposing would be considered a home occupation, as long as the activity doesn't occupy more than 25% of your residence or more than 900 sq feet in an accessory structure. Home occupations are allowed anyplace in town, and no notification or permit is required. Clearly, as others have mentioned, there are plenty of places with more restrictive regs. The only way to know for sure is to look into your local ordinances. If you're inclined to go ahead despite what you find, read the ordinances yourself rather than tipping off your codes officer by asking questions.
A bigger problem can be restrictive covenants in your deed, so look there as well.