The US has at-will employment and as a result most employees have no separate employment contract1. Legal requirements and responsibilities between employer and employee are defined at both the federal and the state level. They can additionally vary based on the size of the company, its involvement in cross-state trade or government contracts, whether it is a government agency and whether it qualifies as a faith-based organization to name just a few things.
Company policies, whether signed or unsigned, may additionally have some legal power but generally not as much as a contract which is a separate legal concept.
What constitutes the contract legally binding me and my employer?
For most purposes, nothing. You are free to stop coming into work at any moment. An employer is free to fire you at any point for any reason (not specifically related to a protected class). Notice periods are a standard practice but, absent an actual contract, are not enforceable.
This briefly covers the concept of an "employment contract" as it relates to most US employees. For anything specific to your situation, consult a lawyer.
1 One source that I'll care to cite is Alison Green at Ask A Manager: ...unless you have a contract, which most U.S. workers don’t