Yes. Carefully remove the HDD from the old, defective laptop.
Find out, what interface this HDD has (IDE or SATA) then go ahead and buy an IDE(or Sata) to USB adapter (example: Delock 61391).
and put into other laptop and than just turn it on.
Connect the old HDD to the USB adapter you bought and hook the assembly to an USB port of your new computer. If the HDD is not damaged, then it should show up in "My Computer" as additional drive, and you should be able to access the data.