normally for apache to be able to write to a directory this directory has to be writable+executable by the user the apache process runs as, look at the output of 'ps' on unix, for example
ps -Ao user,group,command will give you username,groupname and commands of all running commands on your machine. If your apache runs as user 'nobody', you can make a directory writable+executable by nobody and apache should be able to write there. So:
mkdir -p /some/dir
chown nobody /some/dir
chmod 700 /some/dir
Will create a dir that is exclusively readable/writable by 'nobody'. The 'read' permission is not strictly necessary but normally is quite handy.