My hamster is lethargic, losing hair, and has a neck bump. Cancer?

Poor Bear! From the picture, I can see that he has a flat area which looks like it has been scratched. Though this can be a sign of cancer, I do not see an obvious lump. This appears to be self-trauma from excessive scratching. Unfortunately this is a vicious cycle, with chronic damage to the skin, Bear is set up for bacterial and fungal infections. While I cannot determine the underlying cause, this can be easily done by your veterinarian, and treatment is typically very effective. Underlying causes can be fur mites, which are very tiny and cannot be seen by the naked eye, allergies, hormonal issues, or a primary skin infection. To make Bear more comfortable before he is seen by a vet, you can carefully trim the nails on his hind feet so that he minimizes the trauma and damage he is creating if he is scratching himself. A silver-sulfadiazene cream or triple antibiotic cream can also be applied to the area once or twice a day.