If you didn't want to pay upfront for Paragon there wasn't much of a chance you would have been able to get into the recent closed beta. The Epic Games team has planned an open beta for all players, however, and it looks as if we're finally getting details on exactly when the open beta will pan out.
The upcoming open beta will start on April 28 and will run until May 1. This upcoming four day Paragon weekend will be the first phase for the open beta, whilst a second phase will spark up open beta play a week later and will continue till May 8. Servers will open up on both April 28 and May 5 at 9AM Pacific Time.
If you already signed up for beta access in the past, chances are that you should be able to play during the first phase, although it's not absolutely guaranteed. Epic Games mentioned in their press release that, The first stress test will include invites for the first 500,000 people who signed up for the beta only.
If you haven't signed up yet, you still can by heading to the Paragon website. If you sign up for the beta before the end of the first phase on May 1, then you'll be able to play. Epic Games confirmed that the second phase will include invites for anyone who signs up before May 1. This should include the initial 500,00 players as well.