Your company uses ArcGIS Online. At some point, your company's ArcGIS Online administrator created an Esri Account for you, so that you can sign in on arcgis.com and do cool things with maps.
But one day you try to sign in to My Esri - perhaps to download a new software release - and you get this message:
"Your active account <username> is not enabled for Esri access."
Accompanying this message is the name of your ArcGIS Online administrator.
Note that this message says "Esri access", not "My Esri access". This is because there is a setting on your ArcGIS Online account that, once turned on, allows you to use other Esri sites like GeoNet, the Esri Academy for training, and yes, My Esri.
Contact the ArcGIS Online administrator listed to have them turn on the Allow or Enable Esri Access setting for your account. This setting cannot be set or changed from My Esri, nor can it be changed by the user themselves.
Some ArcGIS Online organizations may not want to allow their accounts to have Esri access, so you may need to create or use another Esri Account in order to access the other Esri sites.
Here are some helpful resources about accounts.
Enable Esri access. This ArcGIS Online documentation has instructions for the administrator on how to change this setting.