Wash your hands, containers, knife, cutting board, food processor, and other tools in hot soapy water. Let air dry. Rinse the carrots and limes in cool water, let air dry.
Using the shred setting on your food processor (or a box grater), grate the carrots and transfer into a bowl. Peel ginger and grate into carrots. You can use a microplane or box grater. Mix well. Zest the limes into carrot mixture. Squeeze lime juice over carrots, extracting as much juice as possible. A hand-held lemon press works great here! Mix well. In a clean saucepan, bring 1 cup of the filtered water to a boil and dissolve the salt into the hot water and allow to cool. Add 1 cup of room temp or cold filtered water to this mixture. Stir with a clean spoon to combine and let cool.
Add the carrot mixture to your clean mason jar, packing them down tightly. Cover with the salt water brine, leaving at least 1/2" of space (you may have extra water leftover).
Place a clean fermentation weight on top of the carrots. All pieces of the carrot mixture must be submerged under water to avoid mold growth. Remember: fermentation happens in anaerobic conditions (aka without oxygen) so it's important to completely submerge the thing you're trying to ferment! Cover the mason jar loosely with a clean towel or place the lid loosely on top of the jar. You want the beneficial bugs to be able to get in there, so don't close it too tight.
Store your jar at room temperature on a dark shelf or out of direct light (if you're covering the lid with a towel, that works too!). Be sure you put it in a place where you won't forget about it.
After 3 days your carrots should be fermented. If you prefer more tanginess, allow them to continue to ferment longer. Be sure to taste using a clean utensil.
When you're satisfied with the flavor, tighten the lid on your mason jar and refrigerate. It should keep about 9 months. Add as a side to all of your favorite meals!