Share better scores in a game with different devices

Hi everyone,
I have a game already programmed and published (android with Android Studio). Currently the game stores the best score locally (SharedPreferences) and with this I have no problem. Now I have proposed to make a ranking of the best scores of different users on different devices and the truth is that I have never done something like this and I do not know what is the best way. Can it be done relatively easily without the need for servers? The procedure would be: consult the table with the 5 best scores, see if the current score is in the best 5, put it in the appropriate position and save the 5 best scores again. I know it may sound easy but then not so easy, so I would really appreciate your help.
Thank you very much in advance!