I store them, whole and unpeeled, in the veggie crisper in the fridge. They seem to lose their crispness and get soft and soggy in 3 - 4 weeks; by 5 weeks they're shriveled and inedible. Maybe it's because of the humidity in the south? My potatoes don't keep for months, either -- they're moldy in 6 weeks, tops.
I just had to throw away the remainder of a bag of sweet potatoes (about a pound) because they were covered in mold after just 2 weeks in my pantry. I stored them in the mesh bag they came in on a shelf with (but not touching) my other non-refrigerated produce (apples, bananas). Am I doing it wrong?
The humidity around your parts would be a factor that I hadn't considered, however fridges are usually fairly dry due to the circulation of cooled air, so storing your carrots in the fridge outside the crisper could help with that. I don't know 100% why but my carrots used to go rubbery too when kept in the crisper. Keeping them on one of the shelves, in a perforated bag, with free air flow solved this though - now they stay crisp for months, until I use them all up.
In this case I'm talking about locally grown carrots from the grocery store, unpeeled and uncut. In fact the reason that I knew that carrots, when properly stored, can last months is because when running that food bank I used to manage, potatoes, carrots, onions and beets were the primary locally grown veggies available through the brutal winters around here.
The section of my Canadian Gardner book on harvesting carrots told me to store the ones I grew this fall in a well ventilated area of the fridge in a perforated bag with a dry paper towel that I should change if it got moist, which is how I dealt with my own carrots, and they kept well too. It suggests storing them in saw dust in a root cellar for long term storage, but I've never needed to do that because I only grew a couple dozen small variety carrots.
I do keep my potatoes in my basement pantry in a milk crate to ensure they stay cool, dry and have good air flow. I have a few of the ones I grew and harvested in September in there, and they're still doing fine. Reading that you store your potatoes near your apples and bananas makes me think this is also related to why you're not getting a longer shelf life out of them. Both fruits, as well as onions if you've got them, give off gas as they ripen which causes things near them to also speed in their processes of ripening and/or spoiling.
Wombat - as for your question about tasty carrots, I'm not sure if it's a preference thing on your part, but I've been told by farmers and gardeners alike that the soil in my region is ideal for growing superior root crops such as carrots and potatoes due to the nutrients in it. This fall was my first crop of my own carrots, and I have to say I hadn't has such sweet tasty carrots in years. Our garden's newly built with fresh new soil, which apparently helps as well. When it comes to my potatoes, they turned out to be totally usable, but fairly bland in flavour. I went for seed potatoes from Walmart last growing season, but I'm thinking that this season I want to go for something a little more fancy in hopes I'll get a better tasting end result.