What kind of system has this problem? For example, Windows, Mac, Linux, or Android?

I most often see this on Windows, so I'll give you a suggestion for that for now:

Clean Reinstall

We use this name, but it's not about removing your settings, it's about making sure the program files are clean (no inconsistent or alien code files). As described below, this process does not disturb your existing settings. Do NOT uninstall Firefox, that's not needed.

(1) Download a fresh installer for Firefox 38.0.5 from https://www.mozilla.org/firefox/all/ to a convenient location. (Scroll down to your preferred language.)

(2) Exit out of Firefox (if applicable).

(3) Rename the program folder, either:

Windows XP and 32-bit Vista/7

C:\Program Files\Mozilla Firefox

to

C:\Program Files\OldFirefox

64-bit Windows Vista/7/8

C:\Program Files (x86)\Mozilla Firefox

to

C:\Program Files (x86)\OldFirefox

(4) Run the installer you downloaded in #1. It should automatically connect to your existing settings.

Does a cleanly reinstalled Firefox start normally?

Note: Some plugins may exist only in that OldFirefox folder. If something essential is missing, look in these folders:

• \OldFirefox\Plugins
• \OldFirefox\browser\plugins

Once the XPCOM problem is solved, if you still cannot load sites, could you try the suggestions in these articles:

• Fix problems connecting to websites after updating Firefox
• Firefox can't load websites but other browsers can