Is it illegal to save a picture from a snap/story of someone on the snapchat app? I just want to save a picture and keep it for myself, I'm not going to share it or put it on internet.
Can the victim take legal action against you?
Given that you do not intend on sharing the picture, which could open up other areas of law, the only applicable laws that I can personally think of would be copyright laws (assuming the content of the picture is not itself illegal).
Presumably, the person who sent you the snapchat took the picture themselves, and so would hold copyright over the picture. In France, while the picture would be protected under authors' rights, private copying is in general legal. Article L122-5 of the Code de la propriété intellectuelle states:
Lorsque l'oeuvre a été divulguée, l'auteur ne peut interdire :
[...] 2° Les copies ou reproductions réalisées à partir d'une source licite et strictement réservées à l'usage privé du copiste et non destinées à une utilisation collective [...]
My rough translation:
Once the work has been disclosed, the author may not prohibit:
[...] 2) The copies or reproductions originating from a lawful source and strictly reserved for the private use of the copier and not intended for collective use [...]
Therefore, I do not believe that any successful legal actions could be brought against you for copyright violations (but I'm also not your lawyer). For more information, the French Wikipedia has an article on private copying (no English wiki equivalent exists as far as I can tell).
As a note to readers in other countries, the legality of private copying varies widely from jurisdiction to jurisdiction and would have to be researched individually for each. Where it is legal, there are often additional taxes on media where copies can typically be made, which gets redistributed to professional creators (including in France).