If you are looking for a direct, standalone download link for ArcObjects SDK 10.8 on a public webpage, it does not exist. Esri has transitioned this specific SDK to "Mature Support" and removed it from their public download portals. You will not find it on the current Esri Downloads page or through a standard Google search.
If you’re looking to download the ArcObjects SDK for 10.8 , you’ve likely noticed that the process isn't as straightforward as clicking a "Download" button on a public webpage. Because ArcObjects is the deep-level customization framework for ArcGIS Desktop (ArcMap), its availability is tied closely to your organization’s licensing. arcobjects sdk 10.8 download
Visual Studio Support: 10.8 officially supports Visual Studio 2017 and Visual Studio 2019. .NET Framework: Version 4.5.2 or later is required. The Hard Truth: No Public Direct Download If
Log in to My Esri: Sign in at the My Esri portal with a global account associated with your organization. Visual Studio Support : 10
To debug into ArcObjects internals (advanced users only):
C:\Program Files (x86)\ArcGIS\DeveloperKit10.8C:\Program Files (x86)\ArcGIS\DeveloperKit10.8\DotNetC:\Program Files (x86)\ArcGIS\DeveloperKit10.8\HelpVisual Studio 2017 or 2019 (Community, Professional, or Enterprise). IDE (for Java)