Sounds to me that your web server could be hit by a DDOS attack, given that you cannot access any of your websites.... It could be the attack is targeting another site on the shared server which is not yours and hence you're feeling the pain too....
Check your server log if you can... If you cant logon onto cpanel then for sure you're under attack (or that its server maintenance but your host should inform you of that)...
A quick fix would be to register with a CDN like [URLnf="https://www.cloudflare.com/"]cloudflare[/URLnf] with using the free account.... The setup will be to point your domain DNS to cloudflare and within Cloudflare to your website... The next step is to set the "Help I'm under attack mode"...
That said you should also contact your webhost and seek their help as the first call of action.... They maybe implementing server updates etc....
Finally if you cant get any adequate support and resolution - I would consider moving to another host, one which has the server resources to support your sites traffic, bandwidth etc...
On that note best of luck!
[EDIT] - Re-read the OP and its clearer what is stated... Fully agree that upgrading is the way to go... Noting the great advice given below.... The highly unlikely exception would be fake traffic ...