One of the best places to store unripe tomatoes is on the counter at room temperature. To speed up ripening, put them in a paper bag. Also, keep them stem-side down. Ripe tomatoes are meant to be eaten within a couple of days, so buy them in small amounts as needed. Keep them at room temperature stem side up, away from sunlight. Room temperature, here refers to a temperature of about 70 degrees Fahrenheit or 21 degrees Celsius.