Let $f_n: [0, 1] \to \mathbb R$ be a sequence of positive functions in $L^\infty$ (hence a fortiori in $L^1$) that are equibounded in $L^\infty$ norm - that is $\sup_{n \in \mathbb N} \|f_n\|_{L_\infty} \leq M$ for some $M > 0$.
Is it true that there exists some absolute positive constant $c < 1$ such that
$$\inf_{n_k} \sup_{i, j > N} \|f_{n_i} - f_{n_j}\|_{L^1} \leq cM$$
for all such sequences $f_n$?
Where the first infimum is taken over all increasing sequences $n_k$ of naturals.
Edit: I improved the constant to $c = \frac{2}{3}$. (Later edit: But the optimal constant turns out to be $c = \frac{1}{2}$, see Yuval Peres' answer.)
Answer: Yes, we have $$ \inf_{(n_k)} \sup_{i,j \in \mathbb{N}} \|f_{n_i} - f_{n_j}\|_{L^1} \le \frac{2}{3} M, \label{1}\tag{$\ast$} $$ for each sequence $(f_n)$ in $(L^1)_+$ whose sup norm is bounded by $M$. So we can choose $c = \frac{2}{3}$.
To see this, let $[0,\mathbf{1}] \subseteq L^\infty$ denote the positive unit ball in $L^\infty$.
Lemma. Three functions $f_1, f_2, f_3 \in [0,\mathbf{1}]$ cannot have mutual $L^1$-distances that are all strictly larger than $\frac{2}{3}$.
Proof. Set $g_1 = |f_1 - f_2|$, $g_2 = |f_1 - f_3|$ and $g_3 = |f_2 - f_3|$. For any three numbers $r_1,r_2,r_3 \in [0,1]$, the sum of their three mutual distances in $\mathbb{R}$ is at most $2$.
Hence, $\int g_1 + \int g_2 + \int g_3 \le 2$, which shows that it can't happen that all three functions $g_k$ have norm strictly larger than $\frac{2}{3}$. $\square$
Proof of the claim. We may, and will, assume that $M=1$. Assume for a contradiction that we can find a sequence $(f_n)$ in $[0,\mathbf{1}]$ such that the infimum in the question is strictly larger than $\frac{2}{3}$.
Then there exists $n_0$ such that $\|f_{n_0} - f_n\|_{L^1} > \frac{2}{3}$ for infinitely many $n$ (otherwise we could recursively construct a subsequence $(f_{n_k})$ such that the supremum in \eqref{1} is no more than $\frac{2}{3}$); let's denote the set of these $n$ by $J$.
For any two $j,k \in J$, it follows from the lemma that $\|f_j - f_k\| \le \frac{2}{3}$. Thus, you can take the elements of $J$ to be the indices of your wanted subsequence $(f_{n_k})$. Contradiction, since we assumed no such subsequence exists. $\square$
Remark. It's easy to see that the constant $\frac{2}{3}$ is optimal for the lemma (divide $[0,1]$ into three distjoint intervals $I_k$ of measure $\frac{1}{3}$ and define $f_k = \mathbf{1} - \mathbf{1}_{I_k}$), but I don't know whether it is optimal for the answer to the question.
The sharp constant is $c=1/2$. As in Jochen's answer, we may and shall assume that $M=1$.
Proposition: Let $f_n: [0, 1] \to \mathbb R$ be a sequence of positive functions in $L^\infty$ such that $\sup_{n \in \mathbb N} \|f_n\|_{L_\infty} \leq 1$. Then
$$\inf_{\{n_k\}} \sup_{i, j \ge 1} \|f_{n_i} - f_{n_j}\|_{L^1} \leq 1/2 \,,$$ where the infimum is over all strictly increasing sequences $\{n_k\}$.
To see that this is sharp, consider the functions $\{b_n\}$ where $b_n(x)$ is the $n$th bit in the binary expansion of $x$. Note that $\|b_n-b_m\|_1=1/2$ for all $n \ne m$.
Lemma 1: For any $k$ numbers $y_1,\ldots ,y_k$ in $[0,1]$, we have $$ \sum_{i=1}^{k-1}\sum_{j=i+1}^k |y_j-y_i| \le \lfloor k/2 \rfloor \cdot \lceil k/2 \rceil \le k^2/4 \,.$$ Proof: We may reorder the $y_i$ so that $y_1 \le y_2\le\ldots \le y_k$. Every interval $[y_\ell,y_{\ell+1}]$ is included in $\ell(k-\ell)$ intervals of the form $[y_i,y_j]$ with $i \le \ell <j$, and $\max_\ell \ell(k-\ell)=\lfloor k/2 \rfloor \cdot \lceil k/2 \rceil$.
Lemma 2. Given $k$ measurable functions $f_1,\ldots, f_k$ taking values in $[0,1]$, there exist $i<j$ so that $$\|f_i-f_j\|_1 \le \frac{ \lfloor k/2 \rfloor \cdot \lceil k/2 \rceil}{{k \choose 2}} \le \frac{k}{2(k-1)} \,.$$.
Proof: By Lemma 1, for each $x \in [0,1]$ we have $$\sum_{i=1}^{k-1}\sum_{j=i+1}^k |f_j(x)-f_i(x)| \le \lfloor k/2 \rfloor \cdot \lceil k/2 \rceil \,, $$ so integrating gives $$\sum_{i=1}^{k-1}\sum_{j=i+1}^k \|f_j-f_i\|_1 \le \lfloor k/2 \rfloor \cdot \lceil k/2 \rceil \,.$$ Since the minimum of ${k \choose 2}$ numbers is at most their average, the claim follows.
Proof of proposition: Let $f_n: [0, 1] \to \mathbb R$ be a sequence of positive functions in $L^\infty$ such that $\sup_{n \in \mathbb N} \|f_n\|_{L_\infty} \leq 1$. It suffices to show that for every $c>1/2$ there is a strictly increasing sequence ${n_k}$ such that
$$ \quad \sup_{i, j \ge 1} \|f_{n_i} - f_{n_j}\|_{L^1} \leq c \,.\label{2}\tag{$\ast$}$$ By changing the values on a set of measure zero, we may assume that each $f_n$ takes values in $[0,1]$. Fix $c>1/2$ and find $k$ such that $\frac{k}{2(k-1)} <c$. Define a graph on the positive integers where there is an edge $\{i,j\}$ iff $\|f_i-f_j\|_1 > c$. By Lemma 2, this graph does not contain a clique of $k$ nodes, so by Ramsey's Theorem [1] there is an infinite independent set (i.e., an anti-clique) in this graph, which proves \eqref{2}.
Remark: Related considerations in Hilbert space are in [2].
[1] https://en.wikipedia.org/wiki/Ramsey%27s_theorem#Infinite_graphs
