Pronoun cases (e.g. "my," "his") before gerund constructions

It stands in the way of my being successful.

It stands in the way of me being successful.

This led to his succeeding in the workplace.

This led to him succeeding in the workplace.

These are all fine.

The personal pronoun subjects of gerund-participial clauses functioning as complement are either genitive case (here, "my/his") or accusative case (here,"me/"him)".

The choice between genitive and non-genitive depends mainly on style: the genitive is characteristic of fairly formal style.