Given information:
- Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job.
- Two machines can finish the job in 13 days.

To find:
- How many men can finish the job in 13 days.

Let's assume that the job requires x days for three men and eight machines to complete it.
According to the given information, the job will take half the time taken by three machines and eight men to finish the same job. So, the time taken by three machines and eight men to finish the job is 2x days.

It is also given that two machines can finish the job in 13 days. So, we can say that two machines can complete the job in 1/13th of the time taken by three machines and eight men.

Using the concept of work done, we can set up the following equation:
(3 men + 8 machines) * x = (3 machines + 8 men) * 2x
Simplifying the equation, we get:
3x + 8x = 6x + 16x
11x = 22x

Dividing both sides by 22x, we get:
11 = 22

This equation is not possible, which means our initial assumption is incorrect. Therefore, we need to find another approach to solve the problem.

Let's assume that the job requires y days for three machines and eight men to complete it.
According to the given information, the job will take half the time taken by three men and eight machines to finish the same job. So, the time taken by three men and eight machines to finish the job is 2y days.

Using the concept of work done, we can set up the following equation:
(3 men + 8 machines) * 2y = (3 machines + 8 men) * y
Simplifying the equation, we get:
6y + 16y = 3y + 8y
22y = 11y

Dividing both sides by 11y, we get:
22 = 11

This equation is also not possible, which means our assumption is incorrect again.

Let's try another approach:
Let's assume that the job requires z days for three men to complete it.
According to the given information, three men can finish the job in half the time taken by three machines and eight men. So, the time taken by three machines and eight men to finish the job is 2z days.

Using the concept of work done, we can set up the following equation:
3 men * z = (3 machines + 8 men) * 2z
Simplifying the equation, we get:
3z = 22z

Dividing both sides by 22z, we get:
3/22 = 1/z

Therefore, we can conclude that the job can be completed by 3/22 of a man in one day.

To find how many men can finish the job in 13 days, we can set up the following equation:
(3/22) * x = 13
Simplifying the equation, we get:
x = (13 * 22) / 3
x = 286/3
x ≈ 95.33

Since we cannot have a fraction of a man, we can round up the value to the nearest