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Each BP runs one or more eos nodes and each node has to have identical blockchain data.
You can run a full node too like anyone else.
For more information I would recommend you do some reading on dev portal.
https://developers.eos.io/

EOS Wiki
https://github.com/EOSIO/eos/wiki

and StackExchange for Q&A
https://eosio.stackexchange.com/

Also there is a EOS Resource Planner you can use to calculate your resource needs.
https://www.eosrp.io/

On the question about where to store data. Currently not yet available, but in development is a "storage" contract for EOS that will allow users to store
off- blockchain data on IPFS based storage servers that BPs can offer.

So that means we still not able to decentralize web , the only benefits of eos is that is fast

Have patience. Storage contract will be finished soon enough.