If the site can be browsed normally from a different network, it means that the local network configuration is blocking Amazon CDN servers.
Solution:
Please white list the CDN IPs to allow browsing Amazon CDNs. The list can be found by following the link at IP Address range used by OnDemand

This still happens on Chrome as well (in week (17 - 20th Nov). It occurs via a 3rd party corporate network, that is not the same company tied to the screenshots above. So unfortunately I can't speak to that company's administrator to ask on their end what their proxies - sorry :/

All other Atlassian Jira pages are fine through the aforementioned network, it's just when trying to access User Management screen, you get the weird screen as per the photo and can't go anywhere except back button.Accessing User Management via home network totally fine.
I may have to close this issue as I will not be able to speak with actual administrator on other network to test out settings. Or could be kept open (or reopened under another user) for others experiencing same issue. Thanks and sorry for inconvenience.

There may be other factors at play here, such as IE's security settings or proxy settings. User Management loads correctly on IE11 for me.

The screenshot was from just a couple of weeks ago. I still see the same effect today - Internet Explorer 11 seems to definitely be the culprit.

User Management uses both aui-cdn (hosted by Amazon CloudFront) and cdnjs (hosted by CloudFlare). aui-cdn is used for the styles, cdnjs is used for the scripts. This has been the case since May 2014.

cdnjs is known to have been leased IP addresses that are being blocked by the Great Firewall of China (see https://github.com/cdnjs/cdnjs/issues/1922). If this is the only problem, one should at least be able to see a page with Atlassian styling but likely no content.

The original screenshots, however, depict a page with no styling whatsoever, which implies that not even the stylesheets from aui-cdn are being loaded. It may be the case that these are just old screenshots from before May where our stylesheets were also being hosted on cdnjs.

I'm still seeing this issue on our Atlassian OnDemand instance, but only in Internet Explorer now - previously it was all browsers, as far as I can tell (I might be wrong on that, though). Anyway, I've attached a screenshot of the JavaScript console error messages that appear on page load, as well as a network trace showing that the resources are being loaded successfully in Internet Explorer 11.0.9600. Hope this helps... I'll just use Chrome for now (not a bad thing).

For what it's worth, id.atlassian.com just looked like this (no CSS) in the same IE browser. Could be related...