This sequent is provable.

\begin{logicproof}{2}
    \lnot (a \lor b) & prem. \\
    a \lor \lnot a & LEM \\
    \begin{subproof}
        a & ass. \\
        a \lor b & $\lor$i1 3 \\
        \bot & $\lnot$e 4,1 \\
        \lnot a \land \lnot b & $\bot$e 5
    \end{subproof}
    \begin{subproof}
        \lnot a & ass. \\
        \begin{subproof}
            b & ass. \\
            a \lor b & $\lor$i2 8 \\
            \bot & $\lnot$e 9,1
        \end{subproof}
        \lnot b & $\lnot$i 8-10 \\
        \lnot a \land \lnot b & $\land$i 7, 11
    \end{subproof}
    \lnot a \land \lnot b & $\lor$e 2, 3-6, 7-12
\end{logicproof}

Alternate solution:

\begin{logicproof}{2}
    \lnot(a \lor b) & prem \\
    \begin{subproof}
        a & ass \\
        a \lor b & $\lor\mathrm{i}_1$ 2 \\
        \bot & $\lnot$e 1,3
    \end{subproof}
    \lnot a & $\lnot$i 2--4 \\
    \begin{subproof}
        b & ass \\
        a \lor b & $\lor\mathrm{i}_2$ 6 \\
        \bot & $\lnot$e 1,7
    \end{subproof}
    \lnot b & $\lnot$i 6--8 \\
    \lnot a \land \lnot b & $\land$i 5,9
\end{logicproof}