I don't know anything about ENS and was wondering does it work if DNS is down or unavailable in your location?
DNS lives on the core DNS servers and is replicated elsewhere. DNS being "down" would mean that the core DNS servers were taken down.
ENS [1] lives on the Ethereum blockchain. ENS being "down" would mean that Ethereum itself was down. ENS names (.eth) resolve to address hashes on the Ethereum blockchain.
So to answer your question, they are parallel systems, so one going down doesn't affect the other.
However, the thing I'm wondering is if DNS goes down, does it take ENS with it? I'm not sure how the whole ".eth" part works, is that on DNS.