Safe Mode does the equivalent of a Disk Utility (Repair Disk) while it puts up that progress bar. It never get more than about 1/3 of the way across, even when everything is great. But if your drive is un-mountable after the repair cycle, your Mac cannot start up, and it will shut down.
NOT shutting down is a good sign.
Safe Mode should continue and load a minimal set of Apple-only Drivers. It does NOT load the graphics card Driver, which is why it can be useful to learn more about your graphics card.
--------
If you are using your Boot SSD for everything, and are not using TRIM, your SSD may be clogging up with deleted data. There are several ways to eliminate or reduce the deleted data.
One way is to run any system, 10.6.8 or later, with TRIM Enabler installed, and run one pass of Disk Utility (Repair Disk) with TRIM active. This presents one additional line of output:
"Trimming unused blocks"
and pauses for up to a minute while the unused blocks are TRIMmed.
TRIM Enabler has become much more difficult to install on 10.9.5 and later, because it conflicts with the new Kernel-extension signing feature.
Edit> If you are really running 10.6.8, you can install TRIM Enabler in your active System. It is free, but be sure to get the correct version for your system. There is an advanced-features version also available for a modest charge.