Only if you have it on an external drive. If the unit is still functional you could try attaching an external drive and transferring it. Then again, the main cause for failure is usually the hard drive.
You have to pay Dish Network a one time $40 fee to use an external hard drive to transfer your shows to it. Maybe you can get them to waive the fee since you want to transfer your shows due to equipment failure so that you can transfer them back to the new receiver in this way.