"I couldn't withdraw money from the ATM" is the most natural way to say it. Using the noun instead of the verb would be, "I couldn't make a withdrawal from the ATM". In this context, "withdraw" has the specific meaning of taking money out of an account.
"Draw out" and "take out" are grammatically correct and you would be understood from the context, but they're not natural language. "Draw out" would never be used when talking about money and accounts, it's always "withdraw".