>I want to upgrade IE 8 to IE 11 or install Chrome on Server 2008 r2. But when I click on Download button in either case, nothing works.
>
>Does IE 8 need to have Javascript enabled (I think it is not enabled) in order to download and install Chrome? (or upgrade to IE 11)?

By default IE on Windows Servers runs in Enhanced Security Configuration (ESC) mode. You (or the sysadmin, if you don't have sufficient privileges) will need to temporarily disable that in order to be able to download anything using IE.

https://4sysops.com/archives/how-to-disable-internet-explorer-enhanced-security-configuration-ie-esc-in-windows-server-2008/

As a general rule it's a bad idea to have any browser installed on a server, as it's an attack surface. That's why IE on a server is in ESC by default. Far better to connect to a workstation and download using that. If you must install a browser, uninstall it when you're finished (you can leave the installer available for re-use later if needed).

IE11 may be available as an optional update in Windows Update, which would be a clean way to get and install it. But, if you're not an admin you may not be able to install it via Windows Update.