Add the ginger, horseradish, onion, garlic, jalapeno and lemon juice/zest to a clean jar. Pack them down lightly so that the jar is about 3/4 full.
Use a fermenting weight to hold down the veggies/roots, or place heavy roots at the top so that they will weigh down the herbs and jalapenos (which float).
Pour a generous amount apple cider vinegar over the roots/vegetables. You want everything to stay under the liquid to prevent spoilage. Keep in mind that some of the roots will expand a little so top it off well.
Put the lid on. (If you're using a metal lid, line it with parchment paper / wax paper so that the vinegar doesn't corrode it.) Place in a dark, room temperature cabinet for 2-4 weeks. (A month is best.)
When the cider is ready, shake well and then strain the roots/veggies using a cheesecloth or mesh strainer, then pour liquid into a clean jar.
If desired, add raw honey to taste. I recommend starting with ¼ cup and adding one tablespoon at a time until it reaches the sweetness level you prefer.