It is a classical result of Barany and Grinberg (generalizing an earlier result of Spencer) that there exist $\lambda_1,\dotsc,\lambda_N\in\{\pm 1\}$ with $$ \|\lambda_1a_1+\dotsb+\lambda_Na_N\| \le 2n. $$ The paper of Barany-Grinberg was published in 1981, but they indicated that the problem goes back to a yearwas posed in 1963 paper by Dvoretzky. I was excited to learn that BaranyInterestingly, Barany and Grinberg, andalong with Gergely Ambrus, have just published another paper on this subject!.
Notice that for $n=2$, it is easy to get the sharp bound $$ \|\lambda_1a_1+\dotsb+\lambda_Na_N\| \le \sqrt 2; $$ it follows by using induction and observing that among any three vectors in $\mathbb R^2$ of length at most $1$, there are two vectors such that either their sum, or their difference has length at most $1$, and that for any two vectors of length at most $1$, either their sum, or their difference has length at most $\sqrt 2$.