If a person has no employment contract that limits the reasons that he can be terminated, then he is considered an employee at-will. The employment at-will doctrine states that either the employer or the employee may end the employment relationship at any time and for any reason.
Over the years, courts and legislatures have carved out narrow exceptions to the employment at-will doctrine for illegal discrimination and retaliation, but they are exactly that -- Exceptions. If the exceptions don't apply, then the courts will simply view the termination as a lawful business decision, and therefore, such a decision will not be overturned even if it was unfair or can be proven to have been poorly made. It sounds harsh, but it's that same principle that allows the employee to immediately quit that job if he were to find something better. In other words, the employer is not shackled to the employment relationship any more than the employee is shackled to it.
With that in mind, your rights are far and few in this situation, unless there was illegal discrimination or retaliation. If you believe so, then please let me know. You didn't mention anything like that, and I don't want to make assumptions like that, so I will assume that's not the case here. Accordingly, the bot***** *****ne is that if you are terminated for failing to handle a job that nobody could feasibly handle under the circumstances, then you'd still have no recourse. I am truly sorry. It sounds like the best option at this point is to find a new job.
I am truly sorry that my answer is bad news for you, but please understand that it would be unfair to you (and unprofessional of me) to provide you with anything less than an honest response. However, if your concerns were not satisfactorily addressed, then please let me know, and I will be happy to clarify my answer.