I was on Minecraft and I realized I forgot to pay for realms, so ran out. Does that mean all of our progress is lost?

For any paid services with storage, they typically have a "grace period" where your data is retained so you can resume said services once you have the required service charges.

This time frame can vary greatly between companies & services. While the Minecraft Realms FAQ's don't specify the exact timeframe, their FAQ's does note that you're able to export your world for a "limited time" after your Realms subscription has expired.

If you renew your Realms subscription within 1 week, your world data & progress is most likely OK.

If you renew your Realms subscription within 2 weeks, you're probably OK as well…

If you renew your Realms subscription within a month, you're in that nasty "Twilight Zone" where your world data MAY be lost.

If you let your Realms subscription lapse for MORE than a month before resubscribing, it's most likely a guarantee that your world data is no more & will need to start a new world from scratch.

In short, time is of the essence when your subscriptions lapse to minimize the risks of data loss.