Can you still get a complete, balanced diet if you have a gluten allergy and are also lactose intolerant?

Certainly: Throughout most of human history gluten-containing grains & dairy products were not consumed (& still usually not in asia). Gluten-containing grains (& other grains) are certainly not a necessary part of a healthy diet, and neither is dairy (though lactose-free dairy products like yogurt are an option). One can get all the nutrients needed from vegetables, fruits, nuts, seeds, beans, eggs & meats.