Can you cast a spell with flashback from exile if it was originally cast from your hand and countered?

Flashback (CR 702.33) lets you cast a card from exile if it has flashback — but only if it is currently sitting in the exile zone. The critical point is how it got there. When a spell is countered, it moves from the stack to the graveyard (CR 701.5a), not to exile. So a countered spell cast from hand ends up in the graveyard, not exile, and flashback cannot apply to it there.

Flashback itself reads: you may cast this card from your graveyard by paying its flashback cost instead. Wait — actually, flashback lets you cast it from your graveyard (CR 702.33a), not from exile. After you cast it using flashback, it is exiled when it would leave the stack (CR 702.33b). So the exile happens as a result of flashback use, not before it.

Putting it together: if your flashback spell is countered after you cast it from the graveyard using flashback, it will be exiled (because flashback's replacement effect still applies — CR 702.33b). But a spell countered before ever using flashback (e.g., countered while cast from hand) goes to the graveyard, where you can still use flashback normally later.

Concrete example: You cast Flashback card Increasing Devotion from your hand; your opponent counters it. It goes to your graveyard. Later you may cast it again from your graveyard by paying its flashback cost. Now if your opponent counters it that time, it is exiled instead of going to the graveyard, and you cannot cast it again.

So being countered from hand does not prevent you from later using flashback — it just means the card goes to the graveyard first, which is exactly where flashback wants it.