How are most ALUs built, and is it possible to 'build your own'?
You can build them completely from basic logic gates, and the result will be a nice piece of art :-).  

The 74xx logic series also contains a 74LS181, a 4-bit slice ALU, which simplifies things drastically. Bit slice ALU's were used to build more complex ALUs (read: longer word lengths), but newer technologies have made this kind of ICs obsolete.
note: TTL (74xx) is just one technology used for logic gates. Rarely used anymore. It was followed by Low-Power Schottky: 74LSxx, strictly speaking also a form of TTL. Nowadays there are dozens of logic families, all based on high-speed CMOS (74HCxx, 74HCTxx, 74ACxx,...)

These days the proper way to create an ALU would be to do it in a CPLD or an FPGA. This gives you a lot of gates, and the HDL (Hardware Description Language) you use to design the ALU is a lot easier and less error-prone than trying to figure out how to make the connections yourself with the logic gates. VHDL and Verilog are the HDLs of the day.  

An alternative method to create an ALU (not using logic gates, though), would be a single parallel EEPROM/Flash. You use inputs A and B and the operation as input (address) and get the result of the operation as output (data). All you have to do is compile the ROM's content, meaning that you have to write at every address what the result of the operation will be for the corresponding inputs A, B and operation. Word size will be limited by the largest ROM size you can find.