One simple idea would be to scale (e.g. with factor 1.15) each of the tetrahedrons of the Sierpinski mesh to make them overlap a little, using:

scaleTetr[scale_][tetr_Tetrahedron] := Module[{center, vects},
  center = RegionCentroid[tetr];
  vects = First[tetr] - Threaded[center];
  Tetrahedron[Threaded@center + scale vects]]

And then:

spm0 = SierpinskiMesh[3, 3];
tetrs = MeshPrimitives[spm0, 3];
scTetrs = CSGRegion["Union", scaleTetr[1.15] /@ tetrs];
spm = DiscretizeRegion[scTetrs]

This can be exported to a .stl-file. Does that work for you?

Hi Rebin,

I am glad that you like my approach. For scaling the final .stl-file try e.g.:

Printout3D[spm, TargetUnits -> Quantity[10, "Centimeters"]]