Remote postdoc in the US legal questions
You are a Russian, with a US social security number and a US bank account, but no current right to work in the US. (These details don't seem to help.) You are considering applying for a position in the US, with work being conducted remotely. You cannot be employed as a US employee, given that you have no right to work in the US.
You can be employed as a remote employee or contracted as a remote company. For the former, the employer would likely need to engage an intermediary to handle payments (unless they have operations in your country of residence). For the latter, you'd need to incorporate a company in your country of residence.
TL;DR: Yes, it is theoretically and practically possible, if the employer is willing to employ/contract you remotely. (I can't comment on the likelihood of an employer's willingness.)