This question was answered by David Aldous, who showed that every infinite dimensional subspace of $L^1$ contains a subspace isomorphic to $\ell_q$ for some $q\in [1,2]$. (It was already known that $L^1$ contains a copy of $\ell_q$ for every $q\in[1,2]$. See his paper Subspaces of $L^1$, via random measures, in volume 267 of Transactions of the AMS.

Soon after Aldous' result, Krivine and Maurey proved a more general result, namely that every stable Banach space contains a copy of some $\ell_p$.

Also, I think that David Garling published (in Lecture Notes in Mathematics?) an account of these results; when I get a spare moment I will come back and update the references with additional information.

$L^1$ contains a copy of $\ell_q$ for every $q\in[1,2]$; I will come back and provide an original reference shortly, however to read about it you probably can't do better than the book Topics in Banach space theory by Albiac and Kalton.

More information in the direction of your question was provided by David Aldous, who showed that every infinite dimensional subspace of $L^1$ contains a subspace isomorphic to $\ell_q$ for some $q\in [1,2]$. Aldous' paper is Subspaces of $L^1$, via random measures, in volume 267 of Transactions of the AMS.

Soon after Aldous' result, Krivine and Maurey proved a more general result, namely that every stable Banach space contains a copy of some $\ell_p$.

Also, I think that David Garling published (in Lecture Notes in Mathematics?) an account of the work of Aldous and of Krivine Maurey; when I get a spare moment I will come back and update the references with additional information.