The system will never boot with out ANY RAM as I quote you said you tried doing, "When I tried booting my computer without any RAM, I got an error code (Beeeeep, Beep Beep; 1 long, 2 short beeps)."
You need to install the proper memory sticks designed for that particular motherboard and try it again.
That board will support up to 2 Gb of 184 pin PC 2700 DDR SDRAM part no. KVR333X64C25/1G memory. I would suggest you use Kingston memory with the SiS661FX chipset. A good place to get 1GB sticks cheap is here; http://www.memorysuppliers.com/ki1gbpcdd6ns.html

Because you bought it as a barebone system that means it came with limited hardware it needs to work properly.
Did it come with a hard drive? CD/RW? Graphics card? Network interface card? Dial up modem? cables to plug in additional hardware? cooling fans needed? CPU heat riser and fan?
Many Barbones systems do not come with these additional parts, some do. I would suggest you do an internal inventory fo wht you do have and verses what you will need before going any further and causing damage to what you do have.

I've had my computer for over seven months, working just fine until the other day. When I was talking about the error codes, I knew the system was not going to POST. However, I did not expect to even get a beeping error code from the mother board since, I thought it was dead. I know that, a regular PC will beep (1 long, 2 short) without memory and, no post.

When I first bought my PC seven months ago, I had originally ordered the wrong CPU thus, NO POST. I returned that CPU and bought the correct one (seven months ago). Now, my computer is acting as if it does not have a CPU.

You did not mention who the motherboard manufacturer is. All I can tell by what I have read so far is that it only supports Penitum 4 800Mhz CPU's, not a Celeron Dual Core. SO I do not know what to tell you

The motherboard manufacturer is ASUStek. The model is P5S800-VM. I was running a SINGLE CORE Celeron D (351) processor. The model line (Vintage) supports Pentium 4 and Celeron. I'm convinced now that it IS the processor. The motherboard is acting as if there was no CPU. But still, I can't determine if it IS the CPU or the socket (motherboard). But, thanks anyway for your help.

Did you try flashing the Bio's to version 1007 on the initial bootup?
http://dlsvr03.asus.com/pub/ASUS/mb/socket775/P5S800-VM/S8VM1023.zip
More than likely this may be your issue. The Bio's out of the box does not support the CPU you are trying to install. To read the steps how to do that properly from the .zip file dowload I have given you above, go here to read the steps you will need to take to create a bootable floppy disk for theh flash utility and update process. http://support.asus.com/technicaldocuments/technicaldocuments.aspx?no=​503&SLanguage=en-us
You can not just throw in any CPU and expect that out of the box the Bio's is going to support it, you do not know how long someone has had that motherboard sitting on a shelf waiting to be sold. And according to Asus that is truly a vintage board and they had many issues with the celeron cpu's that is why there are so many various Bio's versions for them. I mean this board came out back in the days prior to EM64T Technology was even introduced to the public.
Also I have seen in the ppast as a system builder on several of the older Mobo's that when people install the CPU they do not set the cpu in properly and fails to make proper contact. And when they install the heat rizer and fan they actually damage the cpu by forcing the cpu into the socket. This was back in the days before they keyed the cpu and socket its self to only fit in the socket in a particular way or the ziff clamp would not close, and by the looks of this one it may be one of them.