Wash and cut the tomatoes. Check each fruit, cut out any blemishes or moldy spots. Cut the tomatoes lengthwise in quarters.
Put the tomatoes through a strainer – follow the directions for the one you have. I use an attachment for a KitchenAid stand mixer. Extract the most pulp that you can.
When I can tomato paste, I do not use salt. Simply put the tomato pulp in a wide flat non-aluminum (nonreactive) roasting pan so that you can evaporate the liquid. I like these 2.5 inch deep hotel pans, which are easy to clean – the rounded corners help immensely. One of these pans will hold the pulp from 10 pounds of tomatoes – if you have more than that, use at least two pans. (NOTE: a half-bushel of tomatoes weighs about 25 pounds)
Put the pan(s) in a hot oven at 425˚F and leave them to roast for up to 4 hours. Stir occasionally. When you can draw a spatula through the pulp and an empty streak is left behind, and is not filled in again, the tomato paste is ready.
I find a wide flat pan an inch deep in tomato paste to be awkward to handle, so first, I scrape the hot pulp into a bowl or big measuring cup.
Ladle the tomato paste into prepared 4-oz jars, leaving ½ inch headspace. While I prefer the smaller jars, the National Center for Home Food Preservation says you may can tomato paste in (at largest) 8-oz jars. Add citric acid to each jar: ⅛ teaspoon for 4-oz jars, and ¼ tsp for 8-oz (half pint) jars.
Place lids and bands on jars. Process in a boiling water bath 45 minutes (the time is the same for 4-oz and 8-oz jars.)
Let cool, label, remove bands, and store in a dark place.
Notes
I like to take some of my tomato paste and freeze it in a silicon ice cube tray. It's so very handy to have a bit of paste to use as needed. A standard ice cube tray's wells hold roughly 2 tablespoons.