All Mechanical Golems in Unity's deck say the following:
This card cannot be played during your play phase.
It's pretty clear that this means that during your turn, you cannot play a Mechanical Golem card as the card you play after resolving start of turn effects and before using a power (that's the Play Phase).
I presume that this means that you can't also play a Mechanical Golem card through something like Inspired Repair, which lets you play another card (since you would have played Inspired Repair, and resolved its effects, during your Play Phase).
But say that during another player's turn, you are given the opportunity to play a card. For instance, say that Fanatic is defeated and uses her "One player may play a card now" ability. Since it is not happening during Unity's Play Phase, could she play a Mechanical Golem straight from her hand and bypass the regular restriction?