Hamsters sadly only live a couple years and they get all kinds of different problems. If both eyes are buldging out then either something happened traumatically to him or there is a possible lesion/mass in that aspect of the head causing them to buldge.
I would speak to your child about the illness and that the hamster needs to go to the doctor. Discuss that these guys only live a short while. Call your vet, I usually give a break to the parents and take the hamster and euthanize them humanely. I realize the $35 vet visit and $40 euthansia charge is a bit much for a $5 hamster, and most vets will give you a break to have the hamster not suffer.
My prayers are with you during this difficult time.
Dr.V
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Dr. D.A. Vamvakias
Veterinarian
