Older iMac can no longer connect to secure websites

My dad's iMac is about 10 years old running Mavericks. In the last year, he has not been able to connect to his email on Live.com or go to iCloud. These sites give the message, "Safari can't open the page "https://www.icloud.com" because Safari can't establish a secure connection to the server "https://www.icloud.com". For a while he could visit some sites with Firefox or Chrome, but even those aren't working on many sites now. When using the non-Safari browsers, a message pops up saying the browser needs to be updated, however, the browsers don't seem to be updatable any further. Is this a thing with older computers? That they can't be updated enough to visit secure sites? I'm ready to help him buy a new computer, but his needs are so few it seems like a waste of money.