How to load mesh from .obj file using unity script?

Your second approach is correct. Unity considers resources in the resources folder as GameObjects. So you load in the game object of your mesh as you are, then you have to get the component. You forgot to specify the component, so this should work just fine:

GameObject meshObject = Resources.Load<GameObject>("name");
Mesh myMesh = meshObject.GetComponent<Mesh>();

If this is what you are already trying, it is an issue of your asset file that we can't help without more information on where the files is from/how it is made. Try a MeshFilter with this method as well.

As said in this answer you should probably load the asset as a GameObject.

var meshObject = Resources.Load<GameObject>("SilverObject");

Can't confirm it since only on the phone currently - actually I believe this only works for .prefab files.

However, if it works then still Mesh is no component so using GetComponent<Mesh>() won't work.

you can try rather using

var mesh = meshObject.GetComponent<MeshFilter>().mesh;

or also

var mesh = meshObject.GetComponent<MeshFilter().sharedMesh;

instead.

In general you can also use Runtime OBJ Importer in order to load any OBJ + MTL file on runtime

And also note from the official Best Practices for Resources:

Don't use it!

When I took GameObject from the ".obj" file like this:

var obj =  Resources.Load<GameObject>("file");

Then I noticed that in this GameObject (obj) there is only one Transform component and one child GameObject. And if we take this child GameObject:

var childObj = obj.transform.GetChild(0).gameObject;

Then we will see 3 components already: Transform, MeshFilter and MeshRenderer.

Thus, I think that anyone who has encountered a similar problem and cannot take a Mesh from an ".obj" file, this code should help:

var mesh = Resources.Load<GameObject("file").transform.GetChild(0)
                    .GetComponent<MeshFilter>().sharedMesh;