Hi packetlost. Currently the underlying storage engines are memory, and distributed TiKV. For the next beta release (coming this week) we also have a local RocksDB implementation and a distributed FoundationDB storage implementation.
In the long run, we want to implement our own key-value store so that we can offer some of the temporal/versioning aspects (which would allow you to view your data in the graph as it appeared at a particular point in time).
For the browser, SurrealDB sits on top of a key-value store which is powered by IndexedDB, so that SurrealDB can run in the browser, using WebAssembly.