technically, no. one product key per installation of the software is the rule.
HOWEVER, i have never had problems doing this (i've done it maybe thrice). this is because microsoft allows you to install the same software on up to two computers that the purchaser (aka user aka you) uses.
as far as using the software: it should work fine and won't stop working or uninstall itself after a period of use.
I think irain is asking if it's legal for another member of his/her family to use one of the three codes that comes with his/her Office for Mac Student and Teacher's edition.
I haven't read the EULA, but I'm sure it's not legal to reuse a key even yourself. As for another member of the family, my wife, who is a member of the academic staff in UC Berkeley, has this Office for Mac, and I'm using one of keys from it on this iMac.
Of course, technically she owns this iMac and the MacBook that is using another one of the keys.
Either way, MS isn't going to find out and sue you.