You have to wait until the lower pop times of the day. You can see a real time update at http://na.finalfantasyxiv.com/lodest...06d0909189e479

Each servers restriction is updated at 4:00 a.m./ 8:00 a.m./ 1:00 p.m./ 6:00 p.m./ 11:00 p.m. (PDT)

Even the most populated of servers eventually opens up, just need to wait and jump on it early.