Fortunately, a bill is currently being considered by Congress that balances citizens' privacy rights and law enforcement's interest in obtaining evidence for criminal investigations, called the International Communication Privacy Act.
In 2013, federal law enforcement obtained a search warrant to seize the contents of a Microsoft customer's email account as part of a narcotics investigation, pursuant to the Stored Communications Act. Microsoft complied up until the point where the government requested data that was stored wholly overseas. Proceeding under an interpretation of the Electronic Communications Privacy Act, Microsoft was ordered to hand over the overseas data, which they subsequently refused to obey, and were held in contempt.
The ECPA was passed in 1986 before the web was truly "worldwide" and is thus hopelessly outdated when applied to the modern Internet. Most crucially, it allows law enforcement to take possession of any emails or messages without a warrant, so long as they are more than 180 days old.
Generally, exceptions to the Fourth Amendment requirement for a warrant are based on the idea that there is no "reasonable expectation of privacy," which is why the government doesn't need a warrant to search garbage on the curb, for example. In this context, the mere fact that 180 days have passed is an arbitrary expiration date given one's expectation of privacy in an email.