Can a J-1 postdoc start a remote job abroad?

It seems unlikely. This page from Dept. of State discusses adjustments such as change of category, transfer, and among the things noted is that unauthorized employment is grounds for termination. The regulation governing employment says that

(a) An exchange visitor may receive compensation from the sponsor or the sponsor's appropriate designee, such as the host organization, when employment activities are part of the exchange visitor's program.

(b) An exchange visitor who engages in unauthorized employment shall be deemed to be in violation of his or her program status and is subject to termination as a participant in an exchange visitor program.

That would preclude taking another job, other than the current postdoc.