You can buy a vermicomposter or build one. The size of the worm bin will depend on what species or how many worms you are culturing. If you will be raising them outside, your climate and temperatures will impact the productivity of your worms.
Lumbricus terrestris earthworms require cool temperatures (50-70 °F). European earth worms (Eisenia hortensis) can tolerate 40 °F. Red wigglers (Eisenia foetida) are smaller than either and might be a good choice as a whole-prey item as they don't need to be cut into pieces. Below is information on E. foetida specifically, but you may be able to adapt the guide to other species as well.
Red wigglers are the most popular for composting, they are small and tolerate a wide range of temperatures (40-100 °F) but do best between 70-80 °F. If your turtle likes to eat them, they are perfect choice. Make sure they are palatable, as some reptiles, especially garter snakes, do not like them and they may even have toxic properties to them. (Source: http://www.gartersnake.info/articles/2006/feeding-earthworms-red-wigglers-vs-nightcrawlers.php)
You might want to consider two containers, one for breeding and one for gutloading them to make them more nutritious for your turtle.
For breeding your colony, 2 large (64-72 quarts which is around 64 liters) plastic bins can be used: one will have drainage holes and the other will catch the liquid that is produced by general decomposition via worms and microbes. Use substrate such as shredded paper, coco fiber, or peat moss, submerged in water then placed into the bin. Do not use dirt, clay, sand or other soils: no earth for the earthworms. Keep a damp piece of burlap cloth on top to keep moisture in. Aerate the soil once a week by digging material from the bottom up with a blunt tool or your hands. Add water if it feels dry or just add more moist food to the top and make sure to cover with damp cloth. You can dump the liquid in the bottom bin whenever you wish. Pour out into the garden or lawn, it's good fertilizer but a bit smelly.
Here is a good guide on starting your homemade worm bin: http://www.worms4earth.com/getstarted.php
Feed your breeding colony a variety of kitchen vegetable scraps. Avoid meat, dairy and processed salty foods. Worms eat microorganisms growing on the food as well as very tiny bits of the food, so chopped and/or cooked food is best. Do not overload your bin with food, you will get odors and pests. The bedding you use will also be eaten, so you don't have to "feed" too much. Vermicomposters and gardeners have different needs for their worm bins so they can put in a larger variety of plant matter to compost.
For worms you are going to feed out to your turtle, you can put them in a smaller bin (16 quarts which is around 15 liters) with no drainage holes and slightly drier bedding. Put a mixture of tropical fish flakes, oatmeal, and a calcium supplement for 1-2 days before feeding to your turtle. In the wild, earthworms are high in calcium due to minerals in the soil but calcium is lacking in most kitchen scraps. You could also include high-calcium veggies like dandelions or other greens and high-vitamin veggies like squash and sweet potato.
You should be able to harvest some worms within a few weeks, but be sure to leave the biggest ones that are breeding.