Hmmm. I didn't think this would generate such a discussion. Perhaps I need to elaborate a bit:
Classically, assuming the two particles started at rest (hence zero angular momentum of the two-particle system), the two would "keep going back and forth through each other forever". The only...
Electrons are unaffected by the strong nuclear force. They also have zero size (they're point particles). So, yes, they would keep going back and forth through each other forever, assuming they started at rest.