The best way to calculate is to divide the number of possible ways to achieve your four card hand (four aces = only one way) by the total number of possibe four card hands.
Total possible combinations of four cards = 52! / 4! 48! = 52*51*50*49/4*3*2*1 = 270,725.
Using this method, you can expand it to calculate other combinations of four cards. For instance, the probability that your four cards will be Ace of Spades, Ace of Clubs, Ace of Diamonds, and any heart:
Total ways to draw this combination = 13 (One way for the three aces, and thirteen ways to fill the one heart). Probability = 13 in 270725 = one in 20825.