Currently, as I understand they are not. But, there is legislation [1] which proposes extending Fourth Amendment protection to e-mails and communication stored on your behalf on third party servers. I've already written to my Congressperson in support of that bill.
[1] https://www.congress.gov/bill/115th-congress/house-bill/387/
5th amendment is self-incrimination, 4th is unwarranted search and seizure.
I don't know the current law, but it used to be that e-mails left on a server for more than 6 months were treated like abandoned property, which has a lower-bar to clear for the 4th amendment. I seem to recall hearing that this was changed a while back though.
How the 4th amendment applies to any case is complicated though because it's very much a sliding scale of intrusiveness, with higher levels of intrusiveness requiring more cause for searching.
I plan to roll my own email server in the near future, due to legal and privacy concerns. As I understand it, in the US email archives aren't protected under the Fourth Amendment because they're asking for data from a third party, and the Fourth Amendment only applies to self-incrimination.