After submitting my completed short-term resident permit application online, I was having the hardest time on the website selecting an available date for my interview in Ankara. I called the 157 number (which has been a HUGE help and they answer every question patiently as you complete your application) - and they explained you can not register correctly when using a Mac computer. THEN, it still didn't work for me, and so my fiance tried changing the calendar to Turkish instead of English. THEN and only then did it work.
So, wanted to let the community know - even if it says dates are blocked off - they are NOT! There is a systematical error with the calendars translating between languages. Change your calendar to Turkish language (upper right hand corner) and you will find many dates available.
Hope this helps!