If I have a unit and a Zhonya's Hourglass that is a gear-unit at a battlefield. If both units die at the same time, does Zhonya's Hourglass save the other unit?

Yes — the Hourglass gear-unit can save the other unit, and it still dies itself.

While it's a unit at the battlefield, its replacement effect ("If a friendly unit would die, kill this instead. Heal that unit, exhaust it, and recall it.") applies to simultaneous deaths, and a permanent can replace a qualifying event that happens at the same moment it leaves play (Rule 370.4). Both deaths are treated separately and the replacement's controller chooses the order (Rule 373), so you apply the Hourglass to the other unit's death: instead it's killed, and that unit is healed, exhausted and recalled to base. The Hourglass dies to its own lethal damage anyway — no extra cost, since the death was happening regardless.

The one thing you can't do is save the Hourglass itself this way; "kill this instead" is the already-occurring death, so it self-saves only with a second copy of the Hourglass.