Is it possible to say this sentence without "les uns dans les autres"?
If I add "les uns dans les autres", can I add emphasis to the sentence?
In a similar question, I read that "les uns dans les autres" can be changed into "entre-verb", but I don't think "s'emmêler" can be changed to "entre-verb".
If you remove it, it'll be ambiguous as to whether the concern is for the accessories to bunch up each by itself, or for them to get tied with one another.
An example relevant to your sentence would be a bra, because they can get twisted around themselves -- are you concerned with the bras getting bunched up / twisted upon themselves, or about them being jammed together?
In this case you can also use éviter qu'ils s'entremêlent. The construction s'entre- + can be used when there isn't a dedicated verb, but here there's one (s'entremêler) so it would be used rather than * s'entre-emmêler.
Yes, it is possible to say this sentence without "les uns dans les autres".
"s'", from the word "se" means (sort of) themselves, each other. Get where I am going ? If you check this francaisfacile exercise about pronominal verbs, you can see my point and you can verify with the example for a reciprocal pronominal verb:
Ces enfants se battent continuellement (l'un bat l'autre et réciproquement).
which means
These kids continuously fight.
and we don't specify that they fight each other, otherwise the example would be:
Ces enfants se battent continuellement l'un contre l'autre.
