Before Coinbase, there was Bitbank, and sending Bitcoin over emails.
Brian Armstrong got interested in Bitcoin. He was reading a lot about it, and he was tracking events related to Bitcoin. He learned that no Bitcoin online wallet was available then. He and his friend created an Android-based open-source Bitcoin wallet.
Armstrong promoted it on Tumblr, Hacker News, and Reddit (he offered O.1 BTC for a feedback). This got them 15k downloads in 2 weeks plus media coverage.
They also cold emailed people like Garry Tan, stating in the email, that he has received 0.05 BTC. He just needed to create an account to claim it.
A few months later, Armstrong met Ben Reeves from Blockchain.com, they created Coinbase Beta.
It was fully functional and had basic features: transactions, withdrawals and deposits, and checking account balance. Soon, there was a site, a blog, and funds from Y Combinator.