I like him working out.
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In terms of grammatical case, him is marked accusative. If you want to get into syntactic theory, then it is involved in something called Exceptional Case Marking.
This Wikipedia article is a bit confusing, even if you know what ECM is. Essentially, [him working out] is analysed as a clause, so the sentence is structured [I like [him working out]]. So, [him working out] forms a grammatical unit. The Exceptional Case Marking is the fact that him is getting accusative case as a result of the verb in the main clause, even though him is in the subordinate clause (this normally doesn't happen). So him is serving a dual role in this sentence, according to theory.