There are a few ways around this, depending on the use.

The owner of a wallet only cares about transactions relevant to the wallet. From the beginning, Bitcoin has had a method called Simplified Payment Verification (SPV) that uses block headers and only those transactions needed for the wallet to work:

https://bitcoin.org/en/developer-guide#simplified-payment-ve...

This is also described in Satoshi's original white paper:

https://bitcoin.org/bitcoin.pdf

This takes advantage of the fact that a block header uses the Merkle root for the transaction set:

https://www.youtube.com/watch?v=gUwXCt1qkBU

For nodes and miners, efforts to "prune" the block chain have been underway for some time:

https://en.bitcoin.it/wiki/Scalability#Storage