OS X 10.10.3 can't boot past startup screen
I've updated my 2009 iMac to OS X 10.10.3 and it can't be booted to the login screen after that. The progress bar will be stuck at 50% forever on the startup screen — the longest I've waited was about a day and the iMac was really hot as well. I tried getting into Recovery Mode and did a disk repair and it managed to repair 4 permission errors, but there's one permission that persisted and it says (Warning: SUID file "System/Library/CoreServices/RemoteManagement/ARDAgent.app/Contents/MacOS/ARDAg ent" has been modified and will not be repaired). I rebooted again but it still can't boot past the startup screen.
Unfortunately, I can't reformat OS X because my backup copy from Time Machine has been outdated for a few months as I haven't bought another backup drive to continue using it.
One of the options that I could think of is to reinstall another copy of OS X from Recovery Mode, as it is able to retain my data files. But the problem is that my iMac has only 5 GB of free space, and it doesn't allow me to do that. Is there a way to use the Terminal in Recovery Mode to delete some of my apps and files so that I could install a copy of OS X? Or are there any other ways to solve it?
Any help will be greatly appreciated. Thank you.
iMac, OS X Yosemite (10.10.3)