I've never tried to store them myself, but one of the books that I got when I started gardening was the "Encyclopedia of Country Living", which is loaded with information on growing, canning, storing, etc. Its recommendations for globe onions is a follows (minus the typos of my transcribing):
Curing Bulbs. Let them sun-cure for 3 days. Spread them out on top of the ground until tops are thoroughly dry. Then sort. Bulbs that addicentially got cut with the space go into the house to be used soon. Sets that for some reason didn't take off are cured some more (maybe 2 weeks in a shady, airy place) and then saved to be given another chance next year. They often make it the second time they are planted. The winter storage onions are cured some more and then bagged for keeping.
Storing. If properly cured and stored in a dry, airy place, many of your winter-keeper onions will last 4 or even 6 months. Every summer we bring in onions from sets by the gunnysackful for our winter supply. The trick is to sure them very well in the sun for 3 days or so before bringing in. Rip off the "set" portion before drying if you can. The onion grown from a set has 2 parts. One is small and extends up into the a stiff hollow stem. That is the part you should rip off -- or else the onions won't keep long.
After you bring your onions in from the field, continue drying them. We have tried all sorts of systems for this. You can braid them by the dry stems and hang them up in bunches with wire or twine around the bunch. Or them them in an "onion"-type bag or into old pantyhose and hang them up. My current method, and easiest one yet, is to dump them into cardboard boxes and bring them into the kitchen. I check them occassionally by running my hands through the boxes and removing any damp ones. Any onion that feels the least bit damp is on the verge of spoiling. When the rest are absolutely bone-dry from the kitchen's heat (once that old wood cookstove starts going, it really gives them the treatement), they can go under the bed or into the attic -- anywhere they won't freeze and will stay dry.
Slight freezing doesn't hurt them, provided you don't handle them while frozen. Good ventilation is very important to their storage place, and what's why they should be in some sort of loosely woven bag or hanging basket.or braided and hung up or some such, although I get away with a box. They do better stored dry and cool than dry and hot because with too much heat, they have a tendancy to gradually dry away to layer upon layer of papery nothingness as winter goes by.
When we use our big winter onions, we take the largest first and gradually work our way down in size. That way, if there are any left over come spring, they will be small ones suitable to use as sets in the spring planting.
I should also note that I've seen the pantyhose method in use before, I think at my godparent's place. They had loaded an onion into the toe of the stocking, tied a knot, loaded in the next one, tied a knot, etc, so the onions didn't actually touch which might affect their drying out, and then hung them up so they weren't touching anything.
According to the website StillTasty, fresh, raw whole or cut up carrots will last 2 to 3 weeks in the fridge.
I would think that it would be a shorter time (1 week) for shredded carrots. I'd store them in a container with a tight sealing lid, or in a plastic bag with the air squeezed out.
Best Answer
If they seem like they are kind of starchy, you might toss them with a litle toasted sesame oil. The flavor would be appropriate and that will keep them from sticking together. Other than that, just refrigerate like any other leftover. They should be good for about 4 days assuming your fridge is below 40 degrees F, as it should be. Of course they will never be as good as they were the day you made them. For reheating, my first choice would actually be the microwave. A minute at a time, tossing in between. You might need to add a tablespoon of water to help them loosen up.