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:


  
  Do not let fresch chestnuts freeze and thaw (except if you are freezing for a purpose)
  Never store chestnuts in a sealed plastic bag (except dried or frozen)
  Keep chestnuts refrigerated unless drying or sweetening the chestnuts
  If black molds have entered into the chestnut kernel, do not eat the chestnut
  Chestnuts can be stored in the refrigerator for 2 to 3 weeks
  Cooked chestnuts must also be refrigerated if not consumed right away
  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
  Try to purchase fresh chestnuts close to the time you want to eat them