I don't feel safe looking after someone else's stuff.
I can't relax looking after someone else's stuff.
この文脈だと、「落ち着かない」→「To not feel safe, to not be able to relax」
「人のもの」→「Someone else's stuff, someone else's things」
「I don't feel safe looking after someone else's stuff.」、
「I can't relax looking after someone else's stuff.」
どちらでも使えます。
「よかったら郵送しようか?」は、
「Would you like me to send it to you?」
と表現します。