You don't need to work for a company with large datasets to start learning and playing with large datasets. There are freely available repositories of data you can play with. Checkout the AWS public datasets - https://aws.amazon.com/public-datasets/. There should be more then enough there for you to start playing with "big data" tools and getting familiar with how to store, ETL, and query large datasets.
Yes, but when a company asks for experience they aren't thinking in playing with large datasets unless you create an open source tool that becomes useful for a lot of people. Just playing with them wouldn't be suffice.
