First remove the Box and Axes
r=2 Pi; model = ContourPlot3D[Sin[x] Cos[y] + Sin[y] Cos[z] + Sin[z] Cos[x] == 0, {x, -r,r}, {y, -r, r}, {z, -r, r},RegionFunction -> ({x, y, z} \[Function] x^2 + y^2 + z^2 <= r^2),Mesh -> None,PlotTheme -> "ThickSurface",Method -> {"Extrusion" -> .3} , Axes -> False, Boxed -> False] 
Now you might use the answer (thanks @Henrik Schumacher) of your own question from march2020 how to save a file in STL format
Export[…,DiscretizeGraphics[model]] First remove the Box and Axes
model = ContourPlot3D[Sin[x] Cos[y] + Sin[y] Cos[z] + Sin[z] Cos[x] == 0, {x, -r,r}, {y, -r, r}, {z, -r, r},RegionFunction -> ({x, y, z} \[Function] x^2 + y^2 + z^2 <= r^2),Mesh -> None,PlotTheme -> "ThickSurface",Method -> {"Extrusion" -> .3} , Axes -> False, Boxed -> False]
Now you might use the answer (thanks @Henrik Schumacher) of your own question from march2020 how to save a file in STL format
Export[…,DiscretizeGraphics[model]] First remove the Box and Axes
r=2 Pi; model = ContourPlot3D[Sin[x] Cos[y] + Sin[y] Cos[z] + Sin[z] Cos[x] == 0, {x, -r,r}, {y, -r, r}, {z, -r, r},RegionFunction -> ({x, y, z} \[Function] x^2 + y^2 + z^2 <= r^2),Mesh -> None,PlotTheme -> "ThickSurface",Method -> {"Extrusion" -> .3} , Axes -> False, Boxed -> False]
Now you might use the answer (thanks @Henrik Schumacher) of your own question from march2020 how to save a file in STL format
Export[…,DiscretizeGraphics[model]] First remove the Box and Axes
model = ContourPlot3D[Sin[x] Cos[y] + Sin[y] Cos[z] + Sin[z] Cos[x] == 0, {x, -r,r}, {y, -r, r}, {z, -r, r},RegionFunction -> ({x, y, z} \[Function] x^2 + y^2 + z^2 <= r^2),Mesh -> None,PlotTheme -> "ThickSurface",Method -> {"Extrusion" -> .3} , Axes -> False, Boxed -> False]
Now you might use the answer (thanks @Henrik Schumacher) of your own question from march2020 how to save a file in STL format
Export[…,DiscretizeGraphics[model]]