Combine the ginger, sugar, and 1/2 cup of the water in a 2-quart saucepan and set over medium-high heat. Stir until the sugar has dissolved. Remove from the heat, cover, and allow to steep for 1 hour.
Pour the syrup through a fine-mesh strainer set over a bowl, pressing down to get all of the juice out of the mixture. Chill quickly by placing over an ice bath and stirring, or set in the refrigerator, uncovered, until at least room temperature, 68 to 72ºF.
Using a funnel, pour the syrup into a clean 2-liter bottle and add the yeast, lemon juice, and remaining 7 cups of water. Place the cap on the bottle, gently shake to combine and leave the bottle at room temperature for 48 hours. Open and check for desired amount of carbonation. It is important that once you achieve your desired amount of carbonation, refrigerate the ginger ale. Store in the refrigerator for up to 2 weeks, opening the bottle at least once a day to let out excess carbonation.