Is it possible for four consecutive numbers to equal 39?

No.

Any four consecutive numbers must contain two even numbers and two odd numbers.

Adding two odd numbers together always gives an even number.

Adding two even numbers always gives an even number.

To end up with an odd number you need to add and odd number of odd numbers (3 of them, or 5, 7 etc) together.

The nearest ways to get four consecutive numbers to come close to 39 are ...

* 8+9+10+11 = 38

or * 9+10+11+12 = 42