emac start up problem

Some updates can mess things up, but which and in what way is unpredictable.

As you don't have your work backed up, now would be a good time to do this.
Your videos, photos, music, your Mail folder located at
yourusername/Library/ Mail

and this little file,
yourusername/Library/Preferences/ com.apple.mail.plist

which holds your Mail account setup details. Restoring this file to the same place in a new installation will save you setting your Mail account up again from scratch.

And anything else of yours you want to keep, of course.

There is no need to keep applications, they can be reinstalled from the DVD and then updated later.

Copying to a USB stick would be the easiest though expensive option. Also it would be good to have the USB stick formatted a Mac friendly format of HFS+. Disk Utility will do this.
The alternative is burning to CD or DVD's.
And while on the subject of your data, if the hard drive becomes too full this can lead to malfunction. Your eMac has either an 80 or a 160 GB hard disk which can fill up over time, especially if you save a lot of video.
Determine how much free space you have. Either look at the bottom of an open Finder window, or do a Get Info after highlighting the hard disk are two ways to tell the space available. If you find that your disk is full up, copying out and then deleting those files (and emptying the trash) to make space would be worth pursuing.

If lack of space is not a problem, then if you can satisfactorily copy all of your stuff onto external media, then you have the new option to Erase and Install.

would having space issues make it so my computer won't load??

I'm not sure about that specifically, but in general it is something to avoid and worth paying attention to, even now.

Why your Mac will load in Safe mode but not normally seems to be something to do with extra installed fonts, kernel extensions or startup items.
Whilst obtuse the following link alludes to this.
http://support.apple.com/kb/HT1564?viewlocale=en_US

If it was me I would back up the data and do an erase and install. There is little other option really.

How much free space is there on the hard drive? Or more usefully, how much free space was there before you did the archive & install?

If the system disk of a Unix-based computer, which includes Mac OS X, fills up past a certain point, it will block the computer from starting normally. (This is especially "fun" with a 6-foot tall SGI Irix server, but I digress.) The Unix OS requires scratch space on the hard drive for various logs and caches. I don't know if anyone ever determined an exact number for OS X, but if you drop below 500 MB or son free space on the hard drive, you'll end up in trouble.

More: an archive & install (A&I) puts a new copy of the OS on the hard drive without removing the old one. This is normally a good thing, since it lets you drag over any user-installed preference panes, user preferences, etc. (when doing so, drag only a few items at a time, restarting after each batch, to test if it was some user add-on that caused you to do an archive & install in the first place). An A&I, however, takes up about 5 GB of disk space. That could drive a nearly-full hard drive into trouble.

And: when your eMac froze while watching a streaming TV show, that would have left the cache behind since the viewing application (your browser, I'd hazard) wouldn't have been able to clear the cache on closing.

Can you still start in Safe Mode? (Safe Mode loads fewer components than a normal startup and logically should have a smaller scratch disk space footprint.) If you can, you can check just how much free space is left on the hard drive. Anything below 5 GB calls for clearing stuff out. Start with clearing your browser cache. I fusing Safari, select Reset Safari from the Safari menu. If using Firefox, open Preferences, Advanced, Network and click the Clear Now button.

If you have access to another OS X Mac, you could try using the second Mac as the Host and the eMac as the target in FireWire target disk mode.

If nothing else is available or working, you'll need to start up in single-user mode. When single-user mode
startup is complete and you’re at the command line prompt (the # symbol), it’s a prudent precaution to first check the disk for any minor damage using the Unix file system check utility fsck by typing in and entering
/sbin/fsck -fy
Note that Unix commands are case sensitive, and spaces matter. Repeat fsck until it says your drive is OK.

After starting up in single-user mode, you'll need to put the hard drive in read-write mode rather than the default read-only mode: at the command line prompt, type
/sbin/mount -uw /
then press return.

If the system will boot into basic single-user mode, then after the fsck and mount commands , you can attempt to boot further by typing and entering:
sh /etc/rc
This clears out a lot of temporary stuff, including reducing the memory swap files to just one.

You can check how much space if available by typing and entering:
df -h
The first two lines will have the info you want; e.g.,
Filesystem Size Used Avail Capacity Mounted on
/dev/disk0s2 298G 126G 171G 42% /

To exit single-user mode and restart the Mac normally; enter
reboot

Post back if you need to use single-user mode --- there are steps you can take, but if you're not familiar with the Unix command line, the steps are kind of overwhelming (it's kind of overwhelming even if you have used the command line before).