The browser you want is Internet Explorer.
(I know, right? But I'm serious!)
IE lets you turn off tabbed browsing completely with a simple settings change. The following steps are how you do it in Windows 7/IE 10, they may vary slightly for other versions of Windows or IE.
1) Go to Control Panel > Internet Options.
2) Go to the "General" tab.
3) Click the button labeled "Tabs".
4) In the dialog box that pops up, titled "Tabbed Browsing Settings," un-check the box labeled "Enable Tabbed Browsing."
5) Click "OK."
That's it, you now have your 1998-style tabless browser! Enjoy.
But smacktoward, I'm not on Windows! What do I do for a tabless browser on other platforms?
Despair, mostly. You can turn off most tabbed-browsing features in Firefox (go to Options > Tabs and un-check everything), but all that does is suppress things like new windows opening in new tabs; you can still manually open new tabs if you want to. AFAIK Chrome has no way to turn off or suppress tabbed browsing at all.
But come on, you're looking for a real Clinton-era browsing experience! Chrome didn't even exist back then, and nobody used Macs. So spin up a Windows VM, un-tabify IE, pop in a Third Eye Blind CD and go to town.