This is a topology issue.
The mesh is not at all optimized or even anywhere near qualified to be SubSurf'd
Looks like a result of non-traditional polymodeling techniques like Booleans or maybe exported from a freeform modeling application like Fusion360.
Ideally, you'd want your mesh to be of all quads with good flow to get a decent result with SubSurf. I won't go too into it right here, there's a hundred resources online if you want to read into this more.
This mesh is a mess in that regard. A lot of nasty spiders!
So, what's the solution?
Selecting everything and pressing ALT+J to convert Tris to Quads?
No, because that will probably still not fix the fundamental issue with the mesh.
Only thing that would be a proper "answer" would be Manual Retopology, unfortunately. Try looking into BSurfaces or RetopoMT, read up on topology for SubD modeling and play around with it, then post your model and we can work it out together.
Have fun modeling :D