A single LED has a forward voltage drop of between 2 to 3.5V, depending on color. To power 250 of them in a single series string, it would take a minimum of ~500V, so obviously, you have to wire small groups in series where each group shares one current limiting resistor, and then wire many groups in parallel.
This means you can choose an operating voltage which powers a group, like 15V which might power 5ea 2.2V Leds, leaving 4V across the current limiting resistor. Obviously, it would take 25 groups in parallel to get 250 Leds.
Assuming you plan to operate the groups at 20mA, this means that the total current through 25 groups would be 1/2A. At 1/2A, your proposed AA batteries would last about 1 min!!! Try a 12V car battery.