To access or use the blockchain you don't require the entire copy of blockchain. i.e. the copy of the ledger (or bockchain) is not maintained on each connected node, only few special nodes(mining and full nodes) has to have the copy of blockchain. Others retrieve only content specific to them, which is usually taken care by the wallet software.
There are projects such as Cardano (ADA) which address some of the issues of blockchain, but are still experimental. For example, the ever growing size could be solved by pruning the chain.