How should large companies store credentials for important systems, such as root access to an integral machine (e.g. high-traffic web server), or social media accounts?
First, root access to a machine does not require to be shared. The number of human beings that need to be administrators should be small before the number of users that the machine can accept. Said differently, each administrator of a machine should have personal credentials for that machine, that should allow him/her to require administrative priviledges. That is the only way to be able to know who is responsable for what in case of trouble. And that comes out of the box for Unix-Linux systems as well as Microsoft ones.
In the opposite it is no harm in having the same credentials allow access to a group of machines. Single Sign On systems can be found everywhere, from the good old Kerberos to Microsoft Active Directory or LDAP. What matters is that the group of machines must be identified. Even inside a datacenter, it is common for administators to share same credentials for all machines of one security zone.
That does not mean that companies pretending to be serious never do security horrors like installing the credentials for a bunch of machines (mainly machines belonging to their clients) in some relay host. Nor does that mean that credentials for the company facebook account is not written on a paperboard in the communication service office, but those are definitely poor security practices.
In short; most likely (hopefully) some kind of password manager would be used here. As mentioned by Neil Smithline there are tons of ways of actually doing this and most ways are quite awful.
The only "good" method is a password manager of some kind, whether it be KeePass or perhaps an online password manager such as LastPass or DashLane. Realistically it doesn't matter what password manager I give you as an example, ideally, you need to research what would suit your requirements (if you do have a specific problem to solve) these are just examples of what you could use. I am in no way advising you do use them or ill-advising them.
However, as mentioned this is a broad question & it's quite hard to answer as it stands.