For the former, I'd recommend using Amazon's Glacier or Google's Nearline storage.
If you intend to retrieve your files often, Nearline will do better as the costs are lower and retrieval is faster. You'll likely want to wait until it's out of beta first though, which might take a while.
If you don't intend to retrieve them often, Amazon's Glacier might be a good choice.
There are also services like CrashPlan and Backblaze that are definitely worth looking into. They're definitely easier to use and CrashPlan at least gives you the option to encrypt.
For the latter use case, I'd recommend Google Drive. You can get a ton of storage through Apps for Work Unlimited and there's very little chance of it disappearing in the near future with Google's apparent goal of moving all your data into their cloud.