This page offers information on how to obtain the Golden Net.
Completing the Bug Critterpedia
To obtain a DIY for the Golden Net, you’ll need to complete the bug section of the critterpedia. There are a total of 80 Bugs to be caught, and insects are seasonal so it will take you a while to obtain all bugs if you do not time travel or visit different islands in the opposite hemisphere. Donating the bugs is not required to unlock the golden net DIY, but Blathers will be very grateful if you do donate the first of each bug species you catch! When you’ve caught all the bugs, you’ll be sent a DIY in the mail!
Golden Net DIY
To craft the Golden Net, you need a net and a golden nugget. The net required is made from a flimsy net and an iron nugget, and the flimsy net is made from 5 branches. Gold Nuggets are obtained from boulders but they are rare and will not spawn that frequently.
The Golden Net is far more durable than the other nets in the game, however, it is not indestructable. Other than its increased durability, it does not have any special powers.