What size capacitor needed to power solenoid?

I guess you mean capacitor in air conditioner that help to start three phase motor when
it plug on one phase?
Afraid its not good, 1-10 uF what I've seen so far.
250 000 uF should not be costly, as you need low voltage, let say 16V, and
it even better if 5-10 smaller one connected in parallel.