First check how the system believes / is supposed to be mounted:
mount|grep -i " / "
Should look something like this:
/dev/mapper/VolGroup000-LogVol00 on / type ext3 (rw)
Might also be worth running a "cat /etc/fstab" and take a look to make sure there isn't a read-only option set in field four of the / entry.
If it is supposed to be (rw), then this should work:
mount -o remount /
If it is being intentionally mounted (ro), then try this:
mount -o remount,rw /
If that appears to work, make sure it survives a reboot (after checking fstab again).
Failing that, reboot in single user mode and run fsck on the disk in question to check and repair the Linux file system. Invoke fsck without parameters and it should check everything listed in /etc/fstab.