I am a now retired, there is work to be done on the house. If I climb a ladder and fall, could I sustain fractured neck?

Possible. You could very well sutain a serious injury. Why do you not consider asking for help from your church and your community. It is not worth the risk to fall. Also if you have home owners insurance, look at your policy to see if there is anything that could be covered.
Yes... ...Though from a short ladder, hip and upper extremity fractures are more likely. Neck fractures happen mostly when people land head first, and you need more height for that.
Of course. You could also fall down stairs and get hit while crossing the street. If you are in good health and careful then climbing a ladder should be a safe endevour.