While Esri does have some freely available tools, the ArcObjects SDK (.NET or Java) is not one of them. ArcObjects is a licensed product, available from the same distribution mechanism as the rest of the ArcGIS platform.
In general, the procedure for gaining access to Esri software is:
- Contact an Esri representative (via web site, US regional office or international distributor, as appropriate)
- Register a login on the Esri global account system
- License the software (there are a multitude of options here, including non-commercial and educational licenses) -- the licensed software will be associated with a Global Account
- Connect to the My Esri Customer Service portal with your Esri login
- Navigate to "My Organizations" -> "Downloads" and identify the product and version you want to download, then press the "View Downloads" button (ArcObjects is actually accessible through both the "ArcGIS for Server" and "ArcGIS for Desktop" products)
- Review the list of files, and press "Download" for the software package(s) you need (don't forget any ancillary products, like 64-bit Background Geoprocessing, or Additional Products like database client drivers)
- Execute the Windows extraction utility, which will create a folder with a Windows "Setup.exe" executable (Linux and Unix media are often GNUzipped tarfiles and generic resources may be distributed as ISO files).
- Execute the setup utility (assuming you have local access rights to do so)
If you do not have Customer Service access rights, you'll need to find someone in your organization that does, and have them either download the media kit(s) for you (then start at step 7) or ask them to have Esri add your global login as an approved download user for the organization (and start at step 4).