A 16 bit DAC will cost you maybe a few pounds, dollars or Euros. A 16 bit R-2R DAC made at home will not be monotonic and so will cause horrible distortion. A string DAC made at home will be monotonic but not necessarily linear, unless you build it with Vishay S series resistors, of which you will need 65535, all of one value, at about 7 pounds/euros/dollars each, and at least 4096 16 bit analogue multiplexers plus decoding logic.
You CAN make a PWM DAC at home, and it can be highly accurate and linear, but it will need time resolution of about 350 picoseconds to cover the audio band with 16 bit resolution. You would need a FPGA with a phenomenal clock speed and minimal jitter. It will also need a well-designed low pass filter, which is entirely feasible as it involves op amps and only moderately precise (say 1%) resistors and capacitors, and only a handful of each.
A PWM DAC is supremely useful for low frequency instrumentation, where the superb linearity probably can't be beaten, and is a technique worth learning about, but is just not going to be useful for audio.
So basically you can't beat, or come anywhere near to, a moderately priced chip from any of the major semiconductor manufacturers.