The Elder Scrolls Online is now allowing everyone to participate in a closed beta. Those interested can find out how to sign-up for the beta at elderscrollsonline.com.
The sign-up form is rather extensive, but be sure to fill in as much as you can; those who fill in more information have better chances of getting into the beta.
The form is rather extensive and asks a lot from volunteers; it's likely many applicants won't be selected for the beta. It requests detailed information about items such as your computer diagnostics and your past gaming experiences with MMOs and Elder Scrolls games. In some categories, it even asks about your past playing styles.
Those who complete and submit the form will be contacted via email within 48 hours.