I tried to do a backup from within my custom recovery (CWM or TWRP) but it failed due to a lack of space. Since then, my phone will not complete the boot process but remain on the boot animation for hours. (In case it matters: it's not a boot-loop in the strict sense as the phone does not go through the boot process over and over again. It's just stuck and doesn't go all the way through.)
I have tried wiping dalvic-cache and cache. I removed the memory card (to see if it had gotten corrupted and was blocking the boot process). I even re-flashed my ROM and GApps, but to no avail.
Since I did not flash anything or really make any changes to the phone at all, I am stubbornly refusing to believe (as many posts and videos on the net seem to suggest) that the only way to proceed in this situation is to do a factory reset (or some other method like an odin flash) which will delete all my precious data. There must be a way to get this phone going again without losing any data.