Prepare the papayas by washing them with cool, clean water and set aside to drain.
Prepare and heat the pint or quart jars you will use for canning in the oven or hot water.
Peel the papayas, then cut in half lengthwise and scoop out the seeds. Discard the peelings and seeds.
Cut the papaya into ½" to 1" cubes.
In a large saucepan, combine the water and sugar and stir to dissolve.
Cook the sugar water over medium-high heat, stirring occasionally, until boiling.
Add the papaya cubes to the boiling sugar water.
Reduce heat to low and simmer for 2 to 3 minutes.
Add 1 ½ teaspoons of lemon juice to hot, sterilized pint jars or 1 tablespoon of lemon juice to hot, sterilized quart jars.
Using a slotted spoon, pack the papayas into the hot jars with lemon juice, leaving ½" headspace at the top of the jar.
Using a canning funnel, pour the hot sugar water syrup the papayas were cooked in over the papayas, leaving ½" headspace at the top of the jars.
Use a rubber spatula to remove any air bubbles from the jar, and add more syrup to the jar as needed to maintain the ½" headspace.
Wipe the rims of the jars with a clean cloth and apply the 2-part canning lids.
Process the papayas using a hot water bath canning method. Once the water has returned to a full boil, process pint jars for 15 minutes and quart jars for 20 minutes, adjusting time if necessary for altitude. (See notes)
After processing, turn off the heat and leave jars in the canner, uncovered, for 5 minutes. Remove the jars from the canner and let them cool for 12-24 hours, then check that the jars are sealed. Refrigerate any unsealed jars for immediate use.
Sealed papayas will retain peak quality for up to 18 months when stored in a cool, dry place or pantry. Refrigerate once opened.