To check if a chain with $n$ links can be "folded" into a size at most $L$
The crux of the problem is to identify when we can extend a chain $t, s_1a_1, \cdots, s_ka_k$ to a longer chain?
Let $m$ be the length of that chain, i.e., $m= t +\sum_{i=1}^ks_ia_i$. If $m+a_{k+1}\in [0,L]$, then we can extend it with $+a_{k+1}$. If $m-a_{k+1}\in [0,L]$, then we can extend it with $-a_{k+1}$. Otherwise, the chain cannot be extended.
An approach by dynamic programming.
Create a $(n+1)$-row by $(L+1)$-column table $F$ (shorthand for foldable). All entries of $F$ are False initially. $F[k][j]$ will be true if there is a chain made from $a_1,\cdots, a_k$ with length $j$ and within space $L$.

Set $F[0][j]=\text{True}$ for all $j$, $0\le j\le L$. According to the analysis above, $F[k+1][j]$ is true in two cases.

$j-a_{k+1}\in [0,L]$ and $F[k][j-a_{k+1}]$ true.
$j+a_{k+1}\in [0,L]$ and $F[k][j+a_{k+1}]$ true.

In other words, we can fill all values at row $F[k+1]$ if we know all values at row $F[k]$.
If $F[n][j]$ becomes True in the end for any $j$, then we return yes. Otherwise, return no.
Explanation on 5, 1, 7, 2, 8
     0   1   2   3   4   5   6   7   8   9
t 0  T   T   T   T   T   T   T   T   T   T
5 1  T   T   T   T   T   T   T   T   T   T
1 2  T   T   T   T   T   T   T   T   T   T
7 3  T   T   T                   T   T   T
2 4  T       T   T   T   T   T   T       T
8 5  T                           T

The fourth row $F[3]$ corresponds to chains of the form $t$, $\pm5$, $\pm1$, $\pm7$. Their lengths can only be 0, 1, 2, 7, 8, 9. Now if you add $\pm2$ to them, you can get lengths
     2,  3, 4, 9, 10, 11 (for +2)
    -2, -1, 0, 5,  6, 7  (for -2)

So the possible lengths of chains of the form $t$, $\pm5$, $\pm1$, $\pm7$, $\pm2$ are 2, 3, 4, 9, 0, 5, 6, 7, which correspond to row $F[4]$.
How to find the smallest realizable $L$?

A binary search on integers in [a_1, 2M-1] should be a nice way to find the answer, where $M$ is the maximum value of $a_i$s. Binary search is feasible since, if $L$ is realizable, values larger than $L$ are realizable too.
Exercises
Exercise 1. (One minute or less) Show that the smallest realizable $L$ is at most $2M-1$, where $M$ is the largest of $a_i$s.

Exercise 2. Adjust the algorithm if $s_i\in\{-2,-1,1,2\}$ for all $i$ instead of $\{-1,1\}$.