Can you exile a card with flashback from your graveyard during an opponent's turn?

Flashback (CR 702.34) is a static ability that lets you cast a card from your graveyard by paying its flashback cost. The key phrase is 'you may cast this card from your graveyard.' Crucially, flashback does not grant any special timing permission beyond what the card's type already allows — it simply changes the zone from which you cast it (CR 702.34a).

For instants, you may cast them at any time you have priority, including during an opponent's turn (CR 307.1 for instants, CR 116 for priority). So an instant with flashback can absolutely be cast from your graveyard on an opponent's turn.

For sorceries and other non-instant cards with flashback, you can only cast them during your own main phase when the stack is empty and you have priority, unless another effect grants additional timing permissions (CR 307.1). So a sorcery with flashback cannot be cast on an opponent's turn under normal circumstances.

Concrete example: Your opponent attacks you during their combat phase. You have Flashback: {2}{U} on an instant like Think Twice in your graveyard. You may exile it and cast it during that combat phase by paying its flashback cost, since instants can be cast at any time you have priority.

Once cast via flashback, the card is exiled instead of going anywhere else when it leaves the stack (CR 702.34b), whether the spell resolves or is countered.