Looks like you have a DNS problem (and it is highly unlikely that this has anything to do with chess.com).
DNS_PROBE_FINISHED_NO_INTERNET

DNS (Domain Name Resolution Service) is not supplied by chess.com, nor by any other web-site.
It is supplied by your Internet Service Provider... the company that you get Internet access from.
Contact them and complain.

edit: What blueemu says.
And in addition to that it also can be a problem with your computer or router.

DNS queries are resolved, in order of priority, by:
1) your "hosts" file on your local machine
2) if the domain name is not found there, by the ARP table in your local router
3) if the domain name isn't there either, by the designated query server at your internet provider's POS
4) if still not resolved, it is forwarded up the chain to a higher-level server, such as 8.8.8.8 (Google).

perhaps this helps: if you have internet but no DNS, it means you could still use the internet if you knew all the numbers.
Great. But: you can try an alternate DNS server, enter it in your system (google that ), and it would work again.If that does not help, you simply have no internet.
You can tell whether it is a DNS problem or a general internet problem by pinging a site by its number, e.g.
4.2.2.2
so if you enter ping 4.2.2.2 and you get 100% packet loss - no internet.
no packet loss - you need another DNS server but internet is there.
list of DNS servers you can enter as alternates in your system: https://duckduckgo.com/?q=dns+servers
Incidentally, I believe 4.2.2.2 is one.
Good luck :)
DNS_PROBE_FINISHED_NO_INTERNET