That doesn't sound right to me. If there's no air, then only black body radiation can be used to cool the data center. That means a massive radiator, a lot larger than a heat-to-air radiator+fan used on earth.
1. Put data on the moon
2. ???
3. Profit
For more info, check out their promotional video: https://www.lonestarlunar.com/video
Feel free to invest though, perhaps if you feel good about discarding hard drives on the moon I could interest you in space mirrors and for a low low price I'll lease you the spot where your harddrive lands for 100 years.
Okay, so here me out: the ultimate cold storage for someone like Iron Mountain. You'd have to understand that you'll need a minimum of 96 hours for retrieval time, and it's gonna be expensive to get that retrieval rocket there and back. Or, build a big dish and send the data via satellite signals.
You know what they say, 3-2-1 backup: three copies of your data, stored on two different types of media, with one copy kept offsite to guard against common-cause data loss. Depending on the cause, off-site could be a difficult place to go!