I have a 4TB WDSHARESPACE, used to have Mionet to access it from the web. What service can I now use to access it for the net since Mionet is discontinued?

What is avaialble today from WD is the line of My Cloud NAS devices that come in Low price to high price ranges. Storage capacity and features are the main differentiation. You can explore the various types at wdc.com and look at product tab.; the lowest level units are found under Personal Cloud Storage, and the better NAS devices are listed under Network Attached Storage. I advise not getting a “Home” device since there are too many complaints. I would focus on the “true” NAS devices.

I dont want to replace what is currently working, I want to be able to access it like I used to be able to without buying a new NAS. The home cloud units dont give me the security and permissions, for users as far as I can find out, that I currently have

The Mionet service shut down years ago, so it seems to me you are stuck with an obsolete product. Dontact WD xustomer support.