Rinse strawberries under cool water. Discard overripe or damaged fruit. Remove stems and soft spots. Quarter the berries.
Mash berries in batches using a masher or heavy spoon.
Transfer mashed berries to a large pot. Add 1 cup of water. Bring to a boil over medium-high heat.
Reduce to medium-low and simmer for 10 minutes, stirring occasionally.
Remove from heat. Skim foam. Cool for 15 minutes.
Set a fine mesh sieve over a bowl. Ladle in strawberry mixture. Let strain for 30 minutes. Do not press. Discard pulp. Repeat with remaining mixture.
Rinse sieve. Line with 3–4 layers of cheesecloth. Strain juice again into a clean bowl. Discard residue and cheesecloth.
Line cleaned sieve with a coffee filter. Strain juice a third time for clarity.
Measure 4 cups of strained juice into a saucepan. Add ¼ cup strained lemon juice.
Set over medium-low heat. Add 4 cups sugar. Stir until dissolved.
Increase to medium-high. Bring to a boil. Reduce to medium-low and simmer uncovered for 15 minutes, stirring occasionally. Skim foam.
Ladle hot syrup into sterilized half-pint jars, leaving ¼ inch headspace. Wipe rims clean. Apply lids and rings. Tighten to fingertip-tight.
Process in boiling water bath for 10 minutes (15 minutes if above 6,000 feet). For pint jars: 15 minutes (20 minutes if above 6,000 feet).
Turn off heat. Let jars rest in canner for 5 minutes. Remove and cool on towel-lined surface for 12–24 hours.
Check seals. Store sealed jars in a cool, dark place for up to 18 months. Refrigerate after opening. Unsealed jars should be refrigerated and used within 1 month. Syrup can also be frozen up to 6 months.