A lot of people think of "contract" as specifically a written document, but that's not what a "contract" is in law, the written document (if it exists) can be very powerful evidence that (1) there is a contract, and (2) what its terms are, but contracts exist without them.
While US employment is usually at will without a defined contract term, there are mutually enforceable obligations, including some definition of what the employee is obligated to do for the employer and that the employer is obligated to pay the employee at some specified rate assuming the employee's obligations are met. That's a contract. Exactly what the detailed terms are may be difficult to prove absent a single comprehensive written document, but it is a contract.
That being said, "employment contract" colloquially connotes more than "agreement to trade labor for X salary." It implies something other than at-will employment, for one thing.
What good is a contract if you can’t prove what its terms are? Such a contract is worth the paper it’s printed on.
They do print the terms on paper. Usually, companies that don’t have a formal contract that both the employee and employer sign will still write down all the important information. First the employer sends an offer letter containing important information unique to the new employee, such as job title, compensation, work location, start date, etc. Then everything else is in some kind of employee handbook. The handbook details the expectations for every job title, the rules employees are expected to follow, rules for promotions and transfers, etc, etc. Together these have everything you would expect in a contract that both the employee and employer sign, and they are just as binding.