Rinse, dry and rub the fuzz off of the peaches, then cut them into chunks. Place the peaches in a food processor and pulse until you have chunky puree. In a large bowl, combine the peaches with the berries, sugar, and lemon juice, and let them sit for 2 hours.
Fill a large stockpot halfway full with water; this is where you’ll boil the filled jam jars. Insert a canning rack into the pot, or cover the bottom with silverware or balled up tinfoil; you want to create a barrier between the jars and the bottom of the pot, otherwise the jars can crack. Bring the water to a boil.
Wash the jars, lids, and screw bands in hot soapy water and rinse well. Set the screw bands on a clean towel to dry. Place the lids in a small saucepan and cover with water; bring the water to a simmer, then turn off the heat. Place the jars in the stockpot of boiling water, then reduce the heat to a simmer.
Pour the fruit mixture into a large skillet and bring to a boil, then reduce the heat to a simmer and cook, uncovered, for about a half hour. Stir the fruit every few minutes to prevent it from sticking to the bottom of the pan. The fruit will thicken as the liquid in the pan evaporates.
Place a small plate in the freezer. When the jam starts to look thick, place a spoonful on the plate and replace it in the freezer. After 3 minutes, check the jam with your finger or a butter knife; if it has the spreadable consistency of jam, it’s done. If the jam is still loose and watery, it needs to continue cooking, in which case you’ll repeat the same process again in 5 minutes.
When the jam is ready, use a jar lifter to take the jars out of the simmering water. Dump the hot water back into the pot, and shake the jars dry. Turn up the heat and bring the pot of water back up to a boil. Turn on the heat under the pan of lids and bring the water to a simmer.
Pour the jam into the jars, leaving 3/4-inch of space at the top. Slide a chopstick sterilized in boiling water around the inside of the jar to remove air bubbles. Dip a paper towel in boiling water and squeeze it out, then carefully wipe the edges of each jar to remove spattered jam; this is important, as a drop of jam can prevent the jars from sealing properly.
Using tongs, or a magnetic canning lid wand, lift the lids out of the simmering water and shake dry. Place the lids on the jars. Fasten the screw bands around the jars, but not too tightly. Using the jar lifter, place the jars in the boiling water. Add more water if necessary; the jars should have at least 2 inches of water on top. Cover and bring to a rapid boil. Reduce the heat slightly and boil the jars for 10 minutes.
Turn off the heat and wait five minutes, then pull the jars out of the water and place them upright on a dishtowel, cutting board, or flat baking sheet. At this point, you will start to hear the jars create a vacuum seal with a high plink sound. Let the jars cool to room temperatur