Can an employer hire you and then say they can’t because you didn’t pass the background check?

The fact is that most employment is "at will" which means that, as a general rule, a company can set the conditions of work much as it sees fit absent some form of legally actionable discrimination. Therefore, an employer can rescind a job offer based on an applicant's criminal history. That having been said, federal law that prohibits job discrimination. So employees and applicants are protected from policies or practices that among other things disproportionately screen out members of a particular race or ethnicity. Since arrest and incarceration rates are so much higher for African Americans and Latinos, a company/business that adopts a policy of excluding all applicants with a criminal record might be guilty of discrimination. Additionally, since each state has different laws in terms of using an person's criminal history in hiring, you may want to check with your state's department of labor since many of these laws are changing.

Yes, they can do this, unless you had an actual written employment contract guarantying you employment--if you did and the terms of the contract would bar, this, they have to honor the contract. (If they don't, you could sue them for breach of contract.)
However, if you don't have a written employment contract, you are an employee at will. If you are an employee at will, you may be terminated--whih includes having a job offer rescinded--at any time, for any reason, including failing a background check. Basically, without a contract, you have not enforceable rights to a job.