Thoroughly wash the ginger. You don't need to peel it, but break off any hanging pieces as dirt sometimes gets trapped between nodes. Roughly chop the ginger and lemongrass and add it to a food processor along with the lemongrass and 2 cups of water. Process until the mixture is pureed, scraping down the sides a few times with a spatula.
Line a 2 cup liquid measuring cup with multiple layers of cheese cloth, or a single layer of fine-mesh muslin and scoop the ginger mixture onto the cheese cloth; you'll probably need to do this in batches. Gather up the edges of the cloth and twist the edges towards the bulge of ginger to wring out as much liquid as you can. Discard the pulp and repeat until you've squeezed out all the ginger juice. You should now have 4 cups of liquid. If you don't have 4 cups, the ginger you used may have not been fresh, but just add water to make 4 cups.
Add the ginger juice to a large pot along with 6 cups of sugar. Heat the mixture over medium heat to a full simmer. It has a tendency to boil over, so keep a constant eye on it and lower the temperature if it looks like it's going to boil over. The impurities that made the juice cloudy will float to the top as foam, so use a fine mesh skimmer or spoon to remove the foam from the syrup and discard. After the foam stops forming and the syrup is clear, your ginger syrup is done.
You can pour the hot syrup into sterilized glass bottles, or let it cool and transfer it to a suitable container. Stored in the fridge it will last for 6 months.