In other words, we are looking for two positive integers $x$ and $y$ with $x\ge2$, $y\ge2$ and $4\le x+y\le40$ that fit with the conversation.
Let us denote the sum $s=x+y$ and the product $p=xy$.

(1) Danielle's first statement "I don't know what the dimensions of the farm are" gives the following information on the product:

* $p$ is not the product of (exactly) two primes $p=ab$; 
  otherwise Danielle would immediately deduce that $\{x,y\}=\{a,b\}$.
* $p$ does not have any prime factor $f\ge20$; 
  otherwise Danielle would deduce $\{x,y\}=\{f,p/f\}$.

(2) Tammy responded, "I knew you were going to say that Danielle!" 
This means that for any possible way of writing $s$ as the sum of two summands $z$ and $s-z$, the corresponding product does not allow Danielle to deduce $x$ and $y$.
Equivalently, for any integer $z$ with $2\le z\le s/2$, the corresponding product $z\cdot(s-z)$ must be of the form as excluded in (1). 
This excludes most possible values for $s$:

* $25\le s\le 40$: for $z=23$, the product $z\cdot(s-z)$ has been excluded.
* $4\le s\le 24$, and $s$ even:
Then $s$ can be written as the sum of two primes $a$ and $b$ (as a special case of Goldbach's conjecture).
Hence the corresponding product $ab$ was excluded in (1). 
* $5\le s\le 23$, and $s=a+2$ for a prime $a$:
The product of $2$ and $a$ was excluded in (1).

Summarizing we see that $s$ must be a number from the following short list:

> The remaining candidates for the sum $s$ are 11, 17, 23

For convenience, we now enumerate for every remaining sum-candidate the corresponding products that have not been excluded under (1):

* $s=11$: only 18,  24,  28,  30 are possible products.
* $s=17$: only 30,  42,  52,  60,  66,  70,  72 are possible products.
* $s=23$: only 42,  60,  76,  90, 102, 112, 120, 126, 130, 132
are possible products.

(3) Then Danielle said, "Well, I still don't know what the dimensions are!"
Although Danielle knows the product and has deduced the remaining sum-candidates, she is not able to determine the sum. This means that her product $p$ occurrs in the lines of two different sum-candidates.
We update the above listing to these doubly occurring product-candidates:

* $s=11$: only 30 is a possible product.
* $s=17$: only 30,  42,  60 are possible products.
* $s=23$: only 42,  60 are possible products.

(4) So Tammy said, "Well, I know what the dimensions are now!" 
Tammy knows the sum $s$ and the above listing of remaining compatible  product-candidates. If the sum was $17$ or $23$, then Tammy would have been left with at least two candidates and would *not* have known which one was the right one.

>Hence, Tammy must have the sum $s=11$.

(5) And Danielle concluded the conversation by saying, "Then I know what the dimensions are too!" 
Danielle has thought through each of the above steps and has come to the following conclusion:

>The sum is $s=11$ and the product is $p=30$.  
>The dimensions of the farm are 5 and 6 miles, respectively.