Hello. I've not come across this before! It sounds very stressful for you. I would think that it's just the smell rather than anything that could affect a hamster's health. Hamsters do have a very strong sense of smell so it might bother them and cause some stress but I don't think it will make them actually unwell otherwise. Sounds like the main thing is to keep things well ventilated - but that can be tricky if it's very cold where you are? Heating on full and the odd window open a bit maybe?
As for getting rid of smells - I had something similar recently (not skunk but a strong smell on fabric furniture). I tried the usual things, spreading bicarbonate of soda, leaving, hoovering off. Didn't do the trick. Spraying with vinegar (twice) which helped reduce, but not eliminate it. What did the trick in the end was steam cleaning and I actually bought a steam cleaner! Which was not cheap but worked amazingly well.
Is the smell just in the air or on furniture and other items? If it's mainly in the air you could maybe get one of those hepa air purifiers that have a charcoal filter. They run off electricity and are supposed to be very good at removing smells and allergens.
By the way don't use anything like bicarb on the hamster cages or contents as that would be harmful for hamsters.
If the cages and contents have picked up the smell, then the best thing would be to do a full cage clean - and you could use white vinegar to clean the cage - but rinse it off well afterwards. For the contents, freezing or baking should remove aromas. Might be best to wait to do that (if needed) until you've got the smell out of the house generally.
