Your drive being full is not the reason it will not boot, don't listen to these other guys they don't know what they are talking about. Having a full drive is certainly not good for your system and will cause it to slow to a crawl but it will in no way affect the boot process.
I suspect the drive is simply failing from normal wear and tear. Simply take the drive out and put it in a external enclosure like you stated or use the target disk mode like others have suggested. They are in essence doing the same exact thing, mounting the hard drive as an external drive to another computer.
I imagine putting it an external exclosure would give you the best chance of success because then you are directly accessing it with the other computer. I am not that familiar with target disk mode but considering it is communicating through the I/O bus I assume it would require a portion of the kernel to be running on your machine with the bad drive (which may not be able to boot considering the drive is failing). You can try to get it in target disk mode and if it doesn't work then remove the drive and put it in an external enclosure, that will for sure work.
When you start transferring your files off the bad hard drive there is a good chance that it might fail during the copying. If it does don't lose faith and just simply keep trying until you get all the files you need. You should be able to get most if not all of your data intact.
When hard drives start failing they usually don't just die completely. They will start giving bit errors and when that error is detected it will cause the process to fail. Usually the boot files are the first to fail considering they are one of the most used portions of the drive. One time it took me about 10 attempts to make a clone of a failing hard drive (I needed the entire file system intact, not just a few files). It should be significantly easier to get just the files you need (documents, pictures, music, etc.)
Good luck!
