The sum of the showing digits is 139, which is 4 mod 9. 34! is a multiple of 9, so the added digits must total 5, mod 9.

The sum of the odd-positioned digits is 80, and that of the even positions is 59. The difference is 21, one less than a multiple of 11. Each of the two 2-digit hidden areas has its left position in an even position of the number as a whole, and of course the right position in an odd position of the number as a whole. We need to increase the difference to 22 or 33 (or other multiple of 11).

The numbers below 34 include six multiples of 5 including one that's a multiple of 5^2, so the factorial has exactly 7 trailing zeros, making the last of the $'s a zero, but not the next-to-last.

Calling the first pair of $'s AB and the last C0 (that's a zero):

A+B+C could total 5 or 14 or 23.

A and C are in even positions, and increases in them lead to decreases in the difference of 21 in odd-vs-even. B is in an odd position and an increase in that leads to an increase in the difference. At the start they have counted zero toward the sum or difference.

C must be even as 34! has more factors that are 2 than that are 5. Again, it's not zero.

But also, the 2-digit number formed by 5C (concatenation) must be divisible by 4 by the same excess of the prime factor 2 compared to 5. Likewise the last three digits, either 5C0 or 35C must be divisible by 8 due to the plethora of excess 2 as a factor.

     Based on div. by 9              
Value of C       Values of A+B
    2              3, 12
    4              1, 10
    
    6              8, 17
    
    8              6, 15

    

520 and 352 are both divisible by 8, so C=2 is OK.

540 is not divisible by 8, so C=4 is not OK.

356 is not divisible by 8, so C=6 is out.

580 is not divisible by 8, so C=8 is out.

So C must equal 2 and A+B must be equal to 3 or 12, by divisibility-by-9 rules.

    

A + C - B must be congruent to 10 mod 11. 

0 + 2 - 3 works.

1 + 2 - 2 does not.

2 + 2 - 1 does not.

3 + 2 - 0 does not.

If C is 2 and A+B is 3, making B=3 and A=0 results in divisibility by 11.

If A+B were 12, A+2-(12-A)=2A-10,   congruent to 10 mod 11 would require A=10 at least, but A is only one decimal digit. 

So 

AB = 03 and C = 2, and, again, the digit after C is zero.

The result is verified via extended precision:

>> factorial(sym(34))

ans =

295232799039604140847618609643520000000

Also, a calculator could have gotten AB:

 2.95232799039605 x 10^38

 

then merely one of the divisibility rules would have been sufficient to get C.