If I have two units at a battlefield and my opponent plays Elder Dragon, and I react with Flash on both of my units, do they then not take damage from Elder Dragon?

Yes, your units will not take damage in this scenario.

According to the RiftJudge FAQ #9919, Elder Dragon’s play effect has specific targeting restrictions: it allows its controller to choose up to one enemy unit at each board location.

When you respond with Flash to move both of your units from the battlefield to your base:

1. Flash resolves first, moving your units to your base.
2. When Elder Dragon’s ability resolves, it checks the location of its targets.
3. Because the units are no longer at the location where they were originally targeted (the battlefield), they no longer fulfill the targeting restriction of the ability and are unaffected by it.

As stated in the FAQ: "If you play Flash in reaction to the play effect of Elder Dragon, any units that are moved to a location other than the one where they were targeted will no longer fulfill the targeting restriction of the Elder Dragon play effect and will be unaffected by the ability as it resolves."