Prepare a canning pot water bath and three half-pint jars.

Place the tomatoes in a large nonreactive skillet. Stir in the sugar, lemon juice, and vinegar. Let sit, stirring once or twice, until the sugar begins to dissolve, 5 to 10 minutes.

Bring the mixture to a boil over high heat. Reduce the heat to medium high, and cook, stirring frequently, until the tomatoes are soft and the liquid is significantly reduced, thickened, and syrupy looking, 10 to 15 minutes. Stir in the paprika, salt, and cayenne. Continue to cook until the jam has thickened, 2 to 3 minutes more; it is done when you can pull a spatula through the jam and the space you clear stays open for 2 or 3 seconds. The mixture will continue to thicken as it cools, so make sure to stop a little shy of your desired thickness.

Use a funnel to pour the jam into the prepared jars, leaving 1/2-inch headspace. Wipe the rims, apply the lids and rings, and process in a boiling water bath canner for 15 minutes.

Remove the jars, and set them on a folded kitchen towel to cool. When the jars have cooled enough that you can comfortably handle them, check the seals. Sealed jars can be stored at room temperature for up to a year. Any unsealed jars should be refrigerated and used promptly.