We want to find a subspace $F \subseteq \ell_1(\mathbb{R})$ such that for any finite $n \in \mathbb{N}$ and tuple $x=(x_1,\dots,x_n) \in \mathbb{R}^n$ we can find a sequence $(f^k)_k \subseteq F$ with $f^k|_n = (f^k_1,\dots,f^k_n) \to x$ for $k \to \infty$.
Actually, I would expect that this does not exist, but I can't quite find a way to prove this. The usual approach for me would be to say that as $F$ is not dense, there is $h \in \ell^\infty$ orthogonal to $F$, but every initial segment can be approximated by a sequence in $F$ and then try to use a diagonal argument to obtain a contradiction, but I can't make that work.
I hope somebody has some more intuition on that problem than I do.
If a set does not meet some ball, then it cannot be dense in a line through the origin passing through the center of this ball.