The curve network does not meet the requirements for NetworkSrf

From Help for NetworkSrf:

The NetworkSrf command creates a surface from a network of crossing curves.

**All curves in one direction must cross all curves in the other direction and cannot cross each other.**

https://docs.mcneel.com/rhino/7/help/en-us/index.htm#commands/networksrf.htm

That shape cannot be modeled efficiently or effectively modeled as a single NURBS surface in Rhino. (In theory it is possible to model it as a single surface using a large number of control points. However the results would be difficult to modify and could have numerical problems.)

That shape is a good candidate for SubD modeling.

It should be possible to model it as three NURBS surfaces, one for each appendage, connected by blend surfaces. The basic topology of the blend surfaces would be the same as for a Y pipe junction.