Do I need to download all of it locally if I want to develop a dapp?
Is there a website that tracks it in real-time?
I use this for viewing some stats but don't see it here : https://explorer.solana.com/
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It's likely over 100TB now (last I heard a few weeks ago it was in the 90s), you do not need to download it locally and validators don't even keep the entire historical ledger on them.
Unlike other blockchains (e.g. BTC or ETH) the entire history is not required to know the current state of an address or account. This is because with Solana's account model state at any given slot is known, validators store the current state of each account, whereas in BTC for instance you need to run through all historical blocks to work out the current state.
The current size of Solana is in TBs. You don't have to download a full node to develop a Dapp, you can use the public RPC Endpoints provided by Solana