The Kentucky Republican said it took him two hours to complete the process.
"I went through all of my information and then I clicked the sign up button - which I wasn't very happy because of the price - but it didn't sign up and I lost all my data," he told WHAS. "And I had to start all over again and finally got through."
Like other members of Congress, Paul enrolled through the Washington, D.C.-based exchange. The cost for the senator's new health plan is $20,000, about four times what Paul paid for the previous plan covering his wife and three sons, according to WHAS.
This post has been updated.