There is no option on the site to enable this functionality. However, you can do this using userscripts. On Google Chrome, I use Tampermonkey to install and manage userscripts. Firefox users can use Greasemonkey, and there are equivalents for most modern browsers.
The userscript to use is found on StackApps: Display absolute timestamps (in local time). The hosting link is down, but the author, Kip, has posted the code. This may be copied and pasted into a new userscript in a userscript manager of your choice.
Here is a look at the differences in the display with/without the script running: