FPGA LUTs are quite sophisticated, the LUTs in both Xilinx and Intel's latest offerings are able to implement an arbitrary boolean logic function with up to 6 inputs and 1 output. The FPGA also comes packed with other specialized hardware such as on chip memory and multipliers so you do not need to burn logic to use those things. You very likely could fit a Pentium III on a moderately large FPGA.

The real challenge would be matching the clock speed of the original processor. Even 400MHz of the first pentium IIIs on the market might be difficult, getting near 1GHz is likely impossible.