This is a very interesting question, however it seems like a characterization of all subsets of $\mathbb N$ which can be such a degree sequence is very hard. I will point you to the paper "On the degrees of finite extensions of a field" by B. Gordon and E.G. Straus where they study this very problem. Unfortunately, it doesn't seem like the paper is available online so I will include here some of their results. For a field $k$, denote by $S(k)$ the sequence of degrees of irreducible polynomials in $k[x]$, since the only finite $S(k)$ are $\{1\}$ and $\{1,2\}$ by Artin-Schreier we will consider infinite sequences.

• Let $P$ be any set of prime numbers, denote by $N_P$ the set of integers whose prime divisors are in $P$. There is a field $k$ of any prescribed characteristic with $S(k)=N_P$.
• For every field $k$, if $a,b\in S(k)$ and $gcd(a,b)=1$ then $ab\in S(k)$. If $K$ is algebraic over $k$ then every element of $S(K)$ divides some element of $S(k)$.
• For every finite set $A\subset \mathbb{N}$ which doesn't contain $1$, there is a finite $A'\subset \mathbb N$ and a field $k$ so that $S(k)=\mathbb N - A'$ and $A\subset A'$.
• If $S(k)$ consists only of powers of a prime $p$ then $S(k)=N_p$.
• If $k$ is a CE field (all finite extensions are cyclic) then $S(k)$ is either equal to some $N_P$ or to all elements in some $N_P$ which are not divisible by $4$.

There is also something you can say from the perspective of computability. If $P$ is $\Sigma_1$ then there is a field $k$ so that $S(k)=N_P$ and you can take $k$ to be computable. See "Sets of degrees of computable fields".

This is a very interesting question, however it seems like a characterization of all subsets of $\mathbb N$ which can be such a degree sequence is very hard. I will point you to the paper "On the degrees of the finite extensions of a field" by B. Gordon and E.G. Straus where they study this very problem. Unfortunately, it doesn't seem like the paper is available online so I will include here some of their results. For a field $k$, denote by $S(k)$ the sequence of degrees of irreducible polynomials in $k[x]$, since the only finite $S(k)$ are $\{1\}$ and $\{1,2\}$ by Artin–Schreier we will consider infinite sequences.

• Let $P$ be any set of prime numbers, denote by $N_P$ the set of integers whose prime divisors are in $P$. There is a field $k$ of any prescribed characteristic with $S(k)=N_P$.

• For every field $k$, if $a,b\in S(k)$ and $gcd(a,b)=1$ then $ab\in S(k)$. If $K$ is algebraic over $k$ then every element of $S(K)$ divides some element of $S(k)$.

• For every finite set $A\subset \mathbb{N}$ which doesn't contain $1$, there is a finite $A'\subset \mathbb N$ and a field $k$ so that $S(k)=\mathbb N - A'$ and $A\subset A'$.

• If $S(k)$ consists only of powers of a prime $p$ then $S(k)=N_p$.

• If $k$ is a CE field (all finite extensions are cyclic) then $S(k)$ is either equal to some $N_P$ or to all elements in some $N_P$ which are not divisible by $4$.

There is also something you can say from the perspective of computability. If $P$ is $\Sigma_1$ then there is a field $k$ so that $S(k)=N_P$ and you can take $k$ to be computable. See "Sets of degrees of computable fields".