To expand a little on Raghav Sood's answer, Bitcoin miners and many wallets are what is known as full-nodes, they keep a complete copy of the blockchain.
However in the original whitepaper by Satoshi Nakamoto, from which Bitcoin came into existence, Nakamoto identified two ways to reduce the amount of data stored by a node
- "Reclaiming Disk Space" (Pruning)
Once the latest transaction in a coin is buried under enough blocks, the spent transactions before
it can be discarded to save disk space.
- "Simplified Payment verification" (Lightweight wallets)
It is possible to verify payments without running a full network node. A user only needs to keep
a copy of the block headers of the longest proof-of-work chain, which he can get by querying
network nodes until he's convinced he has the longest chain, and obtain the Merkle branch
linking the transaction to the block it's timestamped in
Nakamoto felt that Moore's law and other effects would mean that storage would not become a problem for a very long time.
There is no need to rely on a trusted third-party (cloud or otherwise) to store data.
Wallets also store data locally that is not present in the blockchain. For example: private-keys, indexes, notes attached to transactions etc.