I was reading your post getting more and more of an OMG look on my face, hoping I’d get to the part when you asked if you should have it done. The answer, you should probably have it done.
A few comments though:
First, regarding the problem - yeah, it does sound like the motherboard, but beeps like that can also indicate issues with specific compulsory hardware like the video card, memory, or processor that the system can’t boot up without (as opposed to say a hard drive or CD-ROM failure that would allow at least a limited boot up). Do the beeps have a pattern? Three beeps and a pause? Two beeps and a pause? Etc?
See if you can figure out what kind of motherboard you have, and look it up to see what the beeps could indicate. You can usually find a model number for the motherboard between two of the PCI (expansion) slots. If all else fails, just Google whatever number you can find. 
One other troubleshooting thing to try. Do you know what RAM looks like? It’ll be one or more chips about an inch tall by maybe six inches long installed vertically somewhere on the board. If you only have one chip - nevermind, there’s nothing you can do to test it. However, if there are two or more chips in there, try removing all but one chip, and see if the system will boot up. You can find out how to remove the RAM from the motherboard manual you hopefully found in your earlier googling. 
If that doesn’t work, swap chips and try again. You may get lucky and just have a bad chip, which that procedure would help you discern. If more than one chip doesn’t give you any better results, you can rule that out as a problem.
Regarding taking the whole motherboard out - it’s not rocket science, but it’s certainly not easy. If you’re technically inclined though you may have a shot. Basically you have to remove four things:
- Any expansion cards (like video cards, sound cards, etc)
- The power cable from the power supply
- The array of tiny cables that connect to the motherboard that control the power switch, led lights, and etc.
- Sometimes there will be a small power cable from the power supply for a fan.
If you take meticulous notes (and have the manual handy), I’d think you’d be able to take it out and replace it if you’re mechanically inclined.
Regarding the new motherboard - the mobo doesn’t have anything to do with the speed of your computer (well, technically it does, but likely not in this situation), so any upgrade is likely just going to get you back on your feet. If you’re determined to do this yourself, I’d recommend taking the old motherboard with the RAM and CPU (processor) still attached into a computer store and let the salespeople take a look. They can (hopefully) tell you the type of motherboard you’d need to pick up.
From there, it’s just reading the instructions and hooking it up. Like I mentioned above, there are just those four things to connect to the motherboard, the hardest of which being the little connectors for the power/reset switches, lights, and all. If you do get a new motherboard you’ll need to consult the manual instead of your notes as it’ll likely be laid out differently.
However, if you get it all hooked up and see the wonderful Windows screen appear, the fun still isn’t over. As far as windows is concerned, you’ve just replaced a lot of hardware in one fell swoop, and it’s going to install it all. You’ll see the “found new hardware” screen repeatedly find new stuff on the motherboard and try to install it. It’s very likely that Windows will be very unstable at this point.
There are ways to change the motherboard without reinstalling Windows, but I’ve never attempted it myself. Here’s an article that details it pretty well though:
http://www.pcstats.com/articleview.cfm?articleID=1755
That actually walks through some hardware stuff as well.
Good luck man, I know it sucks being without a computer. That’s why I have a few extras laying around. 