Sorry you are having trouble!
FWIW, the website hasn't been down for literally years. We have monitoring systems and if that were to occur it would be a state of emergency here! You can check this website:
http://www.downforeveryoneorjustme.com/esotericsoftware.com
This can help to see if it is a problem between the routing through the internet between you and the Spine server (it could be your ISP or any of the providers your traffic gets routed through), or if the server is really down.
You can use tracert
on Windows or traceroute
on Linux to show the hops between you and the server, eg:
tracert esotericsoftware.com
This will show at which hop your traffic fails (if it fails).
You can try using this URL to access the website:
http://eu.esotericsoftware.com
This will route your traffic to a datacenter in Europe, then to the datacenter in Japan where the server is. When using this, your traffic between Europe and Japan will be very fast, because it uses the connection between two large datacenters. For example, I am located in Europe and for me it takes 20 hops and 282ms to access the server directly, whereas if I use eu.esotericsoftware.com it only takes 7 hops and 32ms. Though, considering your location based on your IP used to login to the forums, I wouldn't expect you to have problems reaching Japan. Still, this alternate way to access the website may prove useful.
When Spine runs for the first time, it needs to access the internet to download an update. After that, you shouldn't need internet each time Spine starts. Could you please post your spine.log file? The forum will redact your personal information.
Windows: <user home folder>\Spine\spine.log
Mac: <user home folder>/Library/Application Support/Spine/spine.log
Linux: <user home folder>/.spine/spine.log
Can you please try downloading and reinstalling Spine? We've made some improvements today to the launcher that I hope will help.