Prepare your canning setup: Wash and sterilize 6 to 7 half-pint jars. Place new lids in hot water (not boiling), and prepare your water bath canner. Keep everything hot until ready to fill.
Cook the berries to loosen seeds: Combine the mashed blackberries and water in a deep saucepan or jam pot. Bring to a gentle boil and simmer for about 5 minutes, mashing with a potato masher as they cook. The fruit should break down completely and release its juice.
Strain out the seeds: Working in batches, press the softened berries through a fine mesh strainer, chinois, or food mill fitted with a berry screen. You should end up with approximately 6 cups of seedless blackberry pulp. A little more or less is fine—just be sure to adjust the sugar (see notes).
Measure and add sugar: Return the seedless pulp to a clean jam pot. Stir in 4 cups granulated sugar and optional lemon juice. Mix well to dissolve some of the sugar before heating.
Bring to a boil: Turn the heat to medium-high and bring the mixture to a rolling boil, stirring frequently. Use a deep pot, as seedless blackberry jam foams quite a bit.
Cook to gel stage: Boil the jam, stirring constantly, for about 15–20 minutes or until it reaches gel stage. Test for doneness using the plate test, thermometer method, or sheeting test (see notes).
Remove from heat and skim foam: Once gel stage is reached, take the pot off the heat. Skim off any foam from the surface if desired.
Ladle into jars: Carefully ladle the hot jam into prepared jars, leaving ¼ inch headspace. Wipe rims clean, apply lids, and screw on bands until fingertip tight.
Process in a water bath canner: Submerge jars in boiling water with at least 1–2 inches above the lids. Process pints and half pints for 10 minutes, or 15 minutes about 6,000 feet in elevation.
Cool and store: Remove jars and let them sit undisturbed for 12–24 hours. Check seals, label, and store in a cool, dark place for up to 18 months. Refrigerate after opening and use within 2–3 weeks.