If one is hooked up to a cabel modem and the PC is turned on but not on the net can one's PC be accessed by someone else. I have been having problems when I sign on and it is easier to just leave the PC on. I can always unplug the PC to Modem plug but these things have a way of breaking after time.

"can one's PC be accessed by someone else" -> YES

Get a firewall to help prevent access to your pc.

Normally a computer that is 'On' when hooked to a cable or DSL modem is always online, too. There is no login. About the only way to disconnect I know of without unplugging stuff is by going into the Network Status in Control Panel and clicking Disable. You can then re-enable when you want back online. Turning on the 'Show icon in taskbar' in the Properties will make the process a little easier.

Hopefully Jim's advice will work for you. My system is fussy about that (though maybe its my isp). If I disable mine for more than a few hours, it will not find the connection again without a restart. Regardless of if I disable in network settings or with my firewall.

I would also suggest a wired router with built-in hardware firewall. They are cheap. Then you should be able to leave it on with no problems, assuming you have no malware sending info out from your system.