If I use a repeated Rocket Barrage on a Zhonya's Hourglass and a 4 Might unit in their base, does the Zhonya's Hourglass save their unit from the repeated cast?

Whether Zhonya's Hourglass saves the unit depends on how the Repeat instances of Rocket Barrage are resolved.

According to FAQ 5839, when a spell with Repeat deals damage multiple times, the unit is not checked for lethal damage until the entire spell has finished resolving. Zhonya's Hourglass acts as a replacement effect that triggers during the cleanup step after the spell finishes, preventing the unit from dying.

Sequence of Events:

1. Resolution Begins: You play Rocket Barrage and pay the Repeat cost.
2. First Instance: The first instance of Rocket Barrage resolves.
3. Second Instance: The second instance of Rocket Barrage resolves.
4. Spell Finishes: The spell finishes resolving completely.
5. Cleanup/Death Check: The game performs a state-based check for lethal damage.
6. Replacement Effect: If Zhonya's Hourglass is still on the board, it triggers as a replacement effect (Rule 366). Instead of the unit dying, the Zhonya's Hourglass is destroyed, and the unit is healed, exhausted, and recalled to its base.

Because Zhonya's Hourglass is a replacement effect, it does not use the chain. However, if a mode of Rocket Barrage is used to destroy the Zhonya's Hourglass during the resolution of the spell, the Hourglass will be in the trash before the cleanup step and will not be available to save the unit.

Note: I could not fully verify this ruling against the retrieved rules.