Prepare your ginger and turmeric by peeling them. If using fresh turmeric, be mindful that it can stain your hands and surfaces.
Combine the peeled ginger, turmeric (if using fresh), lemon juice, orange juice (if using), cayenne pepper, honey or maple syrup (if using), and filtered water in a high-powered blender.
Blend on high speed until the mixture is completely smooth and liquefied.
Strain the mixture through a fine-mesh sieve or a nut milk bag placed over a bowl or measuring cup. Press down firmly to extract as much liquid as possible.
Discard the pulp or save it for other uses, like adding to smoothies or baked goods.
Carefully pour the strained liquid into small glass bottles or jars. Seal them tightly.