How long do raw chestnuts keep?

According to Chestnuts for Sale:

Chestnut storage is not the same as most nuts. Fresh chestnuts should be stored like carrots. Chestnuts are comprised of about 40-50% water and thus if not stored properly, they will spoil. Therefore, chestnuts should be stored with great care and attention. The ideal storage conditions for chestnuts are 33-35 degrees Fahrenheit and 85-90% humidity. We recommend storing chestnuts in covered containers in the coldest part of the refrigerator. Stored properly in the refrigerator, chestnuts can have a shelf life of approximately 2-3 weeks. If you plan to use the chestnuts soon after receiving them, simply place them in a covered container in the coldest part of the refrigerator. The colder, the better.

Otherwise, they recommend freezing them (preferably par-cooked), and provide information at the link above.

Washington Chestnut offers the following information about storing chestnuts, to prevent molding to which they indicate the chestnuts are susceptible due to their high water content:

1. Do not let fresch chestnuts freeze and thaw (except if you are freezing for a purpose)
2. Never store chestnuts in a sealed plastic bag (except dried or frozen)
3. Keep chestnuts refrigerated unless drying or sweetening the chestnuts
4. If black molds have entered into the chestnut kernel, do not eat the chestnut
5. Chestnuts can be stored in the refrigerator for 2 to 3 weeks
6. Cooked chestnuts must also be refrigerated if not consumed right away
7. Never bathe the chestnuts in plain water - this causes mold spores to be transferred to all the chestnuts passing though the bath water - instead, wash in clean running tap water
8. Try to purchase fresh chestnuts close to the time you want to eat them

I raise and sell fresh chestnuts. The secret of long storage is the proper handling of chestnuts in the first 24 hours after harvest before any deterioration starts to occur. Within that 24 hours our nuts our cleaned, sized, graded and put under refrigeration [32 degrees F.] with controlled humidity [85-90%]. Due to seasonal demand we usually put unsold fresh nuts in the drying room about December 15th for use with dried products the next year. The nuts are still fine at that closing date. Always try to buy farm direct from a US producer. Last year we ate fresh nuts stored our veggie crisper every night until our personal cache ran out which was the first of April. I will have give properly stored nuts a longer shelf life than 2-3 weeks.

Chestnuts spoil in two ways: they go moldy and they dry out and become pretty much impossible to eat when roasted. It's possible that when they are hard and dried out, they could be ground into flour or something; I've never tried that.

Being light is a really good indication that they've dried out. Some chestnuts are moldy when I buy them, and I am pretty sure the ones I bought before Xmas that are still on my counter are either moldy or hard right now. The good news is you can't miss the mold when you take the shells off the cooked chestnuts and you often spot it when you cut the X before you cook them. Don't eat moldy ones. The hardness won't hurt you, so give them a quick bake and see if you actually have any nice ones still (you probably don't.) But don't save them for Valentines!

I find they tend to dry going stale rather than rot, quite quickly too. They will get harder and lighter, and the shells more brittle and apparent gaps between shell and nut, allowing a depression to be made that springs back. The nuts inside shrivel as they dry, which causes that. It's usually immediately obvious when cutting the crosses in the shells for roasting which ones are affected.

When they are like that, they roast very hard and dry and often the shells are hard to remove - not very nice all round, though you could probably grind them still if chestnut flour was the order of the day. Takes a fair while before the flavour is rancid. In short, if you are using them whole or chopped, try to use the heavier ones that feel entirely solid and do not leave them too long. I find the best before dates in the shops are usually within a week or so of purchase and that is not a bad estimate for when they start to get stale.

Place chestnuts in a sink of water...the floaters are the spoiled ones...the good ones are heavier and will sink.

I purchased some last November 2014, I have kept them in the little refrigerator in my office. It cycles between 25 and 35 degrees. Today April ninth I took 2 out to check I peeled both of them there was no black mold or any mold nor are they dried out and gotten hard, I ate one and it tasted good. It appears as though that the temperature rotating between freezing and slightly above freezing has kept them fresh and good.