Hey, you got it right! Blockchain is just an idea for distrubuted database. Hyperledger is an implementation of Blockchain Technology (other examples of Blockchain implementation are:
Bitcoin, Ethereum etc.)
As quoted in Wikipedia:
Hyperledger is an umbrella project of open source blockchains and related tools,
started in December 2015 by the Linux Foundation.
The transactions are stored in the form of blocks in the chain. Every peer in that network has a copy of this. Only the validated and verified transactions are stored in the blockchain.
Yes, Blockchain is a growing set of transactions and as the size of the Blockchain goes on increasing, there would be a problem to store data on every peer because of the increased size. This problem is known as Scalability.