How would you keep an offline-capable app's API data reasonably fresh without losing offline access to it?

Short interview answer

Use network-first or stale-while-revalidate for those requests rather than cache-first: network-first tries live data and only falls back to the last cached response when offline, while stale-while-revalidate shows the cached response instantly and refreshes it in the background, so the app is never blocked on the network but isn't stuck on indefinitely stale data.