BigSur SSH can connect a server in the intranet but cannot connect to a internet server

This is not an Apple problem. I work with literally "thousands" of OpenSSH connections and have not run into such problems. The issue is with the server you are attempting to remote connect. Something is out of date or misconfigured. The fact that changing the port number to 8822 worked, reveals the issue is with the server configuration. That and all your other testing stated that SSH works fine with other server hosts. Digital Ocean might be able to provide some insight with your problem, you could open a support case with them.

If you have an add-on VPN app installed here, remove it, and try again.

Then run an ssh -vvv and compare the results in both paths; non-working and working.

That changing the ports changed behaviors could be the ssh server, or it could also be an issue created by an add-on VPN app.

Compare working old El Capitan Mac vs Newer Big Sur Mac:

/etc/ssh/ssh_config 
/etc/ssh/ssh_config.d/*
~/.ssh/config
~/.ssh/known_hosts
~/.ssh/authorized_keys

As well as any public / private keys

My guess is the server you are attempting to SSH connect to with Big Sur is listening on a different port other than the default port 22 and perhaps your ~/.ssh/config on the El Capitan Mac might have that host configured with the appropriate port. Something like the below config.

~/.ssh/config
Host <customized servername>
  User <user>
  HostName <actual servername w/FQDN>
  Port 2222

You might need to do this on the Digital Ocean host:

sudo ufw allow 22/tcp

You might also need to reboot the Digital Ocean host.

On the Big Sur Mac flush your DNS cache

sudo dscacheutil -flushcache; sudo killall -HUP mDNSResponder

Employer network? Many companies are blocking outbound SSH/SFTP to prevent data loss.

Using telnet to test is all fine and good. But have you compared the configurations between the El Capitan and Big Sur Mac? There has to be a difference because El Capitan is working but Big Sur is failing and in fact timing out seems to indicate it's being ignored and not actively blocked or refused.

Also, in the /etc/ssh there should be several public / private keys and the file permissions need to be proper. I have had issues with Catalina and Big Sur having bad file permissions on these files.

Should be like this (included the octal values so you can "chmod 0660 *_key" and "chmod 0644 *.pub" if you have to alter the permissions to be proper):

0600  .rw-------  1.4k root wheel 27 Apr  2021  ssh_host_dsa_key
0644  .rw-r--r--   590 root wheel 27 Apr  2021  ssh_host_dsa_key.pub
0600  .rw-------   492 root wheel 27 Apr  2021  ssh_host_ecdsa_key
0644  .rw-r--r--   162 root wheel 27 Apr  2021  ssh_host_ecdsa_key.pub
0600  .rw-------   387 root wheel 27 Apr  2021  ssh_host_ed25519_key
0644  .rw-r--r--    82 root wheel 27 Apr  2021  ssh_host_ed25519_key.pub
0600  .rw-------  2.6k root wheel 27 Apr  2021  ssh_host_rsa_key
0644  .rw-r--r--   554 root wheel 27 Apr  2021  ssh_host_rsa_key.pub

You may see other key names in El Capitan and like @etresoft mentioned, it might be an older encryption standard that newer OpenSSH client would reject.

You need to find the difference between El Capitan and Big Sur when it comes to SSH.

You may need to check the servers SSH configuration and update the SSH configuration on your server. Newer OpenSSH versions will be more picky about security. If you are running an old Linux version on Digital Ocean you may need to upgrade it. Depends on what you have configured, without seeing the details, I just don't know. But the problem is on the server somewhere you just need to find it. Or you are missing something that is different on the El Capitan Mac in regards to OpenSSH and its configuration where Big Sur's default configuration doesn't match.

Can you describe "another laptop"? Is this a Mac or a PC? Is it running the same configuration?

Can you describe "a server over the internet"? Do you mean any server, or one specific server?

Have you tried connecting from a different source network?

aaron_ch wrote:

In fact, if I tried my failed macbook to SSH a server in my cooperation network via VPN. It is okay.

Then the problem is on that server. It is probably only allowing some old encryption scheme that is no longer secure, but was acceptable back in El Capitan days.