Your drive is very full! So the OS has no room to work off of.
First Thing To Do: You'll need to make a backup of your files onto an external drive (I would recommend something larger like 3-4 TB). Given the age of your system you'll want to get a FireWire drive as the I/O speed is much faster than USB2. No matter what else do this if you care about all of your stuff and afterwards maintain it.
Second - Options Think about deleting no longer needed stuff (freeing up around 500 MB) or getting a second external drive to use to hold some of your stuff or think about upgrading it to a larger internal HD (2 TB).
The Next Step: You'll need to get a 16 or 32 GB Flash drive to create a boot disk. Then using the Mountain Lion install image prep the flash drive as a bootable drive. Once done you should be able to boot up under it. It will take a bit of time to boot up as the USB I/O is slow. Given how full your HD is now I'm not sure if you could prep a FireWire HD without the installer crashing (it needs a fair amount of memory & HD space to run) which is why we need to do things in this order.
Once the flash drive boots up you can follow what MacHead explained as you need to fix the directory corruption. I would use whats free first, Apple's Disk Utility should get you started here. Then as your drive was so full you'll need to defragment it as well I prefer Drive Genius for it's total collection of tools it offers. What you'll need to do with any 3rd party drive or defragmenter tool is run it from an external bootable drive (in our example here the flash drive). As your current drive is likely one partition.
Replacing your HD: If you decide to upgrade the HD to a larger drive. Make sure it fits your system (height wise). Before you do anything again make a backup of your apps & data, free up as much space as you can and prep the flash drive as outlined above so it's bootable. You will also need to copy over the OS installer to the flash drive as you'll need it to prep your new drive.
Only after you have prepped the flash drive swap your HD out with the new one following the iFixit Guide. Once your drive is installed use the flash drive to boot up from and prep your HD.
I would recommend partitioning it into two partitions. the first being 1/3 of the drive space to hold your OS & apps and have free space for the OS. Then the other partition to hold your files. Doing this lessens the amount of fragmentation as the partition with your data files will fragment out long before the first partition will with just the OS & apps and can be more easily defragmented without needing to boot of an external drive (running from the first partition). This also protects the OS & apps and the needed free space on the first partition.
Lastly, you should try to find a external FireWire case to house your old internal drive so you can use it as a spare drive (USB2 is just too slow IMHO). You'll also need this to copy back your apps and data unless you use the your backup files to recover them. Given the state of your drive currently you may find a few files corrupted hopefully you can rebuild them or re-install the application to fix the damage.
- Good Luck!