I've just tested the thing myself. I have been able to create Clay without any technical problems. So I suppose there shouldn't be a problem for others.
Follow these steps and you should be fine.
-Click "Register".
-You agree with the terms.
-You fill in the form with a valid email address that has been working lately.
-You confirm your password.
-You should get an email with the activation.
-Be sure to check your junk inbox if you haven't received it.
-If you haven't received it, please use another email address. I have no idea how to resend.
-Activate your account and log in with the username and password that you used.
-If they are still problems, check
here to see if you might have problems with cookies.
-Otherwise, something is wrong with your computer.
It worked perfectly on mine. I had to make three accounts.