The basic problem is that your mesh consists of five independent solids which are not correctly put together. This usually happens with tools like Blender or similar 3D-graphic software which has the focus more on nice-looking than on correctness.
Anyway, it's fairly enough to convert your mesh into valid Part solids. The trick is to separate the individual components into own meshes. Currently there is no GUI command that does this automatically. So, what you can do is to copy/paste the original mesh four times to have five versions. Then use the command from "Remove components by hand" and go through the meshes and isolate the corresponding components (i.e. always remove the other four components).
After this is done you can convert all meshes into own Part objects and later convert them to solids. As final step you can do a boolean fusion of the five solids.