There isn't enough space in RAM to store /mach_kernel and the kext cache.

I am sure a PIII can handle OS X.Its just something NextStep derived compiled for Core Duo so its like running i686 compiled Linux on a AMD K6.
Just some Intel 810/815 Mainboards have a RAM limit of 512mb.
Since we don't have full OS X source code on FTP you are limited to Darwin or the compatibility code of NetBSD < 5.x.

*facepalm* Look at the requirements of OSX. It was compiled for an SSE3 i686 target. It's a miracle there's a workaround hackfix for SSE2, it's not going any further. It's nothing like running i686 compiled Linux on an AMD K6. Plus, it would run like crap. Apple's test iMacs for the original intel switch were in fact SSE3 capable Pentium 4s (Prescott core). The kernel was optimized for the extensions that processor had present in it's core.
The bare minimum requirement to run Tiger 10.4 properly is 512MB RAM anyway, and it's not very pretty...
No offence but I suggest you do some reading up on the hackintosh scene and the Intel switch/preparation. There's some useful information you can get from the keynotes.

This man speaks words of wisdom.

There is no chance you'll ever get OSX on that machine. I ran 10.4 on a Pentium 4, 3.02ghz and 768mb RAM. Lets just say, that machine crashed opening up calculator.

I recommend (as many do) Windows 2000 (Runs perfectly on my 233mhz Pentium 2 with 64mb RAM), Windows 98SE, DamnSmallLinux, Xandros (Old Build), or PuppyLinux.

It is not possible to run OSX on P3.... get a P4 (P4 is still not good for OSX86)!
And then atleast install 10.5 to get good software running!
Or go back to Windows 2000...
I recommend Windows Server 2003

Well, OS X Supports less than core 2 duo.
it's Core Duo.

As I said before, it supports Prescott Pentium 4s, at least Tiger did. This unofficial support is because Apple used said processors when they were experimenting before the Intel switch.
Thanks to some OSX86 hackfixes, 10.4 and 10.5 have been known to run on Northwood P4s which only had SSE2. I actually ran 10.5 on a Northwood P4 for a while once.
10.7+ or 10.8+ dropped support for the Core Duo anyway.

Must be 10.8. 10.7 still supports late-2008 MacBooks, which came with C2Ds.

Core Duo, not Core2Duo.
Note the Core Duo is not supported as it is not x86-64 capable.
A bit of research grabs this for 10.7.
Hardware support

The first developer preview of Lion added TRIM support for SSDs shipped with Macs, which is also included in the latest version of Snow Leopard (10.6.8) shipping with current MacBook Pros before July 20, 2011. As of yet, there is no TRIM support for other SSDs.[16]
[edit]System requirements
x86-64 CPU (64-bit Macs, with an Intel Core 2 Duo, Intel Core i3, Intel Core i5, Intel Core i7, or Xeon processor.)[11][17]
At least 2GB of RAM[18]
Mac OS X 10.6.6 or later (Mac OS X 10.6.8 is recommended)[18]
At least 7 GB of free hard drive space[18]
AirDrop is supported on the following Mac models:[18] MacBook Pro (late 2008 or newer), MacBook Air (late 2010 or newer), MacBook (late 2008 or newer), iMac (early 2009 or newer), Mac Mini (mid 2010 or newer), Mac Pro (early 2009 with AirPort Extreme card and mid 2010 or newer).

thx guys got OS X working on this machine.

I would love to know how! That would be a huge breakthrough if you taught us how to do it...

But you were banned, so oh well.

I meant it more theortically than practically.
I have a disassembled SMP PIII-S 1266 system that you can feed up to 4GB RAM (thats much more power and memory than my G4/466)

We could tell users to recompile OS X (impossible because its not really open source) or running Darwin (picky about hardware,no GUI) or use NetBSD 5.x(incomplete emulation scrapped in 6.0)

Interesting to see, that you're also one unsuccessful troll, like some others.

But I think that is the system is kinda slow on this System.
I would at least get a good x86 machine