Some professors a my institution are asking for "common" spaces in which post-docs are not allowed. Is this ok?
You ask if this is common in academia. In all universities I've been in (only Europe), I've never seen something like this. Common rooms where usually for every employee of the respective department (administrative staff included). 

Solar Mike mentions in the comments "Professors have a need to discuss "privileged" information such as grades which postdocs may not have the right to, so separate spaces are a reasonable necessity - and not a social slur...". That professors need to discuss privileged information is true -- but I usually saw a simple solution for that: People were allowed to book rooms for meetings and then only invited people were allowed to enter. This also takes care of the fact that sometimes privileged information needs to be discussed only by some professors (not every professor who drops in), sometimes between professors and postdocs (or between postdocs or phd students). If  your professors' argument is also that they need to discuss privileged information, you might want to suggest this possibility.

However, I suggest you to choose your battles wisely. Academia is a very hierarchical world -- it can be dangerous to start a battle about minor issues (as long as you are not a tenured professors).  Especially if your faculty wants to have a faculty only room (i.e. they themselves are no advocates of "flat hierarchies"), it may be dangerous to suggest too many revolutionary battles. If equality is really so important for you, try to find a professor who advocates for you.

(Depending on how your institution is, you might also want to fight for a "non-professor only" room.)