It turns out that there are 60.  Heres why: 

Case 1:
If we choose 2 or 4 for the last digit, there will be 3 ways remaining
to choose the first digit.  They will be 5,6 or 8  

There are 2 ways to choose the last digit so that there will be 3
ways to choose the first digit. So for each of those 2 ways to choose
the last digit, there are 3 remaining ways to choose the first digit. 
So that's 2x3=6 ways. 

Case 2:
If we choose 6 or 8 for the last digit, there will be only 2 ways remaining
to choose the first digit.  There will be 5 and the remaining one of 6, 8.  

There are 2 ways to choose the last digit (as 6 or 8) so that there will be 
only 2 ways to choose the first digit. So for each of those 2 ways to choose
the last digit, there are 2 remaining ways to choose the first digit. 
So that's 2x2=4 ways.

That's a total of 6+4 or 10 ways to choose the first and last digits.  

For each of those 10 ways to choose the digits on the two ends, we can
choose any of the the remaining 3 digits for the second digit.  That's
10x3 or 30 ways to choose the last, first, and second digits.

For each of those 30 ways to choose the first second and last digits, 
we can choose either of the the remaining 2 digits for the third digit.
That's 30x2 or 60 ways to choose all four digits so that the number
will be even and greater than 5000.

Here is a computer printout of all 60 even numbers
greater than 5000 that can be made with 2,4,6,5,8.


5246 5248 5264 5268 5284 5286 5426 5428 5462 5468 
5482 5486 5624 5628 5642 5648 5682 5684 5824 5826 
5842 5846 5862 5864 6248 6254 6258 6284 6428 6452 
6458 6482 6524 6528 6542 6548 6582 6584 6824 6842 
6852 6854 8246 8254 8256 8264 8426 8452 8456 8462 
8524 8526 8542 8546 8562 8564 8624 8642 8652 8654 

Edwin