I understand that Snapshots are needed in order to make room for new data but it doesn't mean that all data is gone for good, does it?
Is it possible to to access old data if one manages to run a full node wherever a snapshot is stored?
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A snapshot is designed to reduce the size of the tangle to reduce memory burden on nodes. It does this by pruning historical addresses and transactions and saving a baseline of balances at the time of the snapshot.
In order to do this, it prunes the old data, which means from the perspective of the snapshotted node, it is gone.
However, there is no requirement to do a snapshot -- its meant to help efficiency for nodes. What will likely happen is that there will be full nodes that maintain the entire history of the tangle, called permanodes, and all the snapshots that have occurred, with the possibility of operating on a donation or fee-based model to access all of the historical data as an incentive.
Snapshot are publicly available for download. For instance here : https://gist.github.com/paulhandy/5eda521316dec87011128c64722aa20c
Additionnaly, there will be also Permanodes where the full history will be kept.