You might try asking this question on some of the bicycle-specific internet forums such as bikeforums.net or roadbikereview.com.
I found an old copy of the Barnett Bicycle manual and have attached a picture of their page on headset dimensions.
If the headset still works without binding or feeling notches at different steering angles, I would simply remove the stem, unscrew the headset, pull out the fork, clean and re-grease the bearings, then reassemble. Providing you have the tools and expertise to do this successfully. The chances of finding a suitable replacement among modern stock is pretty low. Besides, nowadays, everybody is using threadless headsets which are completely different.