Most of the big players maintain their own datacenters. That said, if you're looking for what the "big boys" do (outside of the top few tech companies), most of them use Amazon S3: https://aws.amazon.com/s3/ (NOTE: I'm not in any way affiliated with Amazon).

S3 is one of the oldest and most popular storage services because:

- It's very fast and can be fronted by Cloudfront, a very large/popular/inexpensive CDN (this is what Netflix uses) - It's inexpensive and gets cheaper with volume - You can choose to use the default redundancy (which is very cost effective) or reduced redundancy which makes it far less expensive - You can archive infrequently used data from S3 to Glacier for a fraction of the cost - It has a ton of features and can be embedded into just about every language/framework in minutes - It's been around forever and has incredible support