Preheat oven to 175 degrees. (200 degrees will also work if your oven doesn’t go that low, though the cook time will be less).
Line a large baking sheet with parchment paper or use the dehydrator rack of your countertop oven.
Cut the tomatoes in half lengthwise and place them cut-side-up on the baking sheet. Sprinkle very lightly with sea salt.
Bake for 2 1/2 hours. Press down on the tomatoes with a spatula to release their juices.
Return to the oven and bake for at least 2 more hours, until tomatoes are dry, keeping an eye on the tomatoes so that they don’t burn.
To Store: Refrigerator: Add the tomatoes to a jar and cover completely with olive oil. Seal and refrigerate for up to 2 weeks. The oil will solidify in the fridge, but once at room temperature, will liquify quickly.
Freezer: For longer storage, keep your tomatoes in the freezer. Allow the tomatoes to freeze in a single layer on a baking sheet. Once they are frozen solid, transfer to a freezer bag and seal. They will keep for up to 3 months.