Exception User-Unhandled

You should check the URL of the service. In the original file it's on line 22:

service.Url = new Uri("https://outlook.office365.com/ews/exchange.asmx");

The error suggests that the application is not able to connect to port 443 on the specified server. When using a custom URL, you should check that an Exchange server is running there and that no firewall is blocking access.

I use exactly the same url in my code.

I mean, I didn't change anything in your code and it still generates that exception... Do you have any ideas where I can check for mistakes?

This is not a problem in the code, it's a network issue. Your computer isn't able to open a TCP/IP connection to the target machine. You'll need to contact your network administrator to resolve the problem. The connection may be blocked by a firewall (intentionally or unintentionally).

I created a Console application in C# and used the same URL in configuration of ExchangeService and it works... The problem occurs only with ASP.NET projects.

It might also be a proxy issue:
https://social.msdn.microsoft.com/Forums/en-US/29a8831a-6251-47e5-90a3-dfe4c87ff20e/systemnetsocketssocketexception-no-connection-could-be-made-because-the-target-machine-actively?forum=netfxnetcom

Yes, thank you so much. I've checked in WireShark the IP address my app was trying to connect with. Then I set that address (with another port) as a proxy and suddenly it started to work. Thank you!

Great, thanks for the update!