Meshing the cow

The cow mesh is an example of a "broken" mesh. Try

mesh =  RepairMesh[mesh]

before sending it to ToElementMesh.

Among other nice meshes, you can find a free and "clean" cow mesh also on Keenan Crane's homepage:

https://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/

This is the model (without texture):

As other's have stated, the issue is self intersecting facets:

mr = RepairMesh[ExampleData[{"Geometry3D", "Cow"}, "MeshRegion"]];

FindMeshDefects[mr]

If we could determine if a point is 'inside' the cow, we could use a naive variant of the powercrust algorithm. Here 'inside' is not necessarily well defined.

Luckily we can use isInside defined specifically for this model here!

dm = DelaunayMesh[MeshCoordinates[mr]];

powercrust = BoundaryMesh @ MeshRegion[
  MeshCoordinates[dm], 
  Pick[MeshCells[dm, 3], isInside /@ PropertyValue[{dm, 3}, MeshCellCentroid]]
];

Needs["NDSolve`FEM`"]

ToElementMesh[powercrust]

ElementMesh[{{-0.410816, 0.410816}, {-0.133851, 0.133851}, {-0.251619, 0.251619}}, {TetrahedronElement["<" 25368 ">"]}]