Some leek varieties, known as "winter leeks", are bred to be slow growing, are very cold tolerant and can be left in the garden over the winter. Other leek varieties, known as "fall leeks" are bred to be fast growing, tender and mild and are not cold tolerant. Winter leeks can just be left in the garden soil, where they will grow slowly until spring when growth speeds up. You can dig them up as needed. If you know the leek variety you can determine whether you have winter or fall leeks.
To harvest and store leeks, dig up the leeks and cool promptly to 32°F. You can place them in crushed ice to lower the temperature to 32°. Leeks are best stored at 32°F under high humidity, which is pretty cold - refrigerator temperatures are 40° or less. Leeks keep well for 10 weeks in a plastic bag under ice in a cooler, which maintains temperatures of 32°. Obviously, it's alot easier to keep leeks in the ground.
The other factor, of course, is what kind of winter we have. This article has additional information on leeks, Leeks https://horticulture.oregonstate.edu/oregon-vegetables/leeks-0.