Cut cored tomatoes into bite-size pieces. Place in a large clean glass jar. (If making a larger amount, use a large glass or ceramic bowl.)
Crush the tomatoes with a clean hand, stir in salt and cover the jar or bowl with a cloth. Secure the cloth with string or a rubber band to prevent debris or critters from landing in the tomatoes.
Stir the tomatoes several times daily. After they become bubbly, continue to ferment and stir for an additional 3 to 5 days. Taste daily. When they taste tangy and have become slightly effervescent, run them through a food mill to remove the skins. Set these skins aside to either kickstart another ferment or add flavor to another dish.
To strain the tomatoes, place a piece of cheesecloth or butter muslin over a jar. With your fist, push the fabric into the jar to create a suspended pouch that you will fill with the tomato pulp and juice. Secure the cloth to the jar with a rubber band or piece of string. Carefully pour in tomato pulp and juice only to the top of the jar opening. Allow the tomatoes to strain a bit before adding more. Place the lid on the jar and the jar on a plate to catch any drips that the cloth may release. Store on the counter or, if straining for more than a day, in the refrigerator. Strain the tomatoes until the paste reaches the desired consistency.
Transfer the tomato paste to a small clean jar and store in the refrigerator. If you won't use the tomato paste within a few days, pour a little olive oil over top to help keep air out of your ferment. Covered in olive oil, the tomato paste will keep for at least a month. Keep in mind however, that as the tomato paste continues to slowly ferment in the refrigerator, the flavor will become more sour.
Remove the fabric from the straining jar that now contains tomato juice, replace the lid or, if desired, bottle the juice. Store in the refrigerator.