How can I browse the web on very old versions of Mac OS?

This answer was last updated in February 2021, and may become outdated as time goes on. As of this writing, however, all software listed is actively maintained with security patches. On OS's which themselves have known vulnerabilities, this is particularly crucial.

Mac OS 8 and below

Attempting to browse the web on these OS's is basically a lost cause. However, you could set up Web Rendering Proxy on another (modern) machine, which will stream websites to older browsers as an interactive gif. If you squint, it's almost like the real thing! https://github.com/tenox7/wrp

Mac OS 9 – Mac OS X 10.3

Also more-or-less a lost cause, but if you want to try your luck, go with Classilla. http://www.floodgap.com/software/classilla/. You might also consider Web Rendering Proxy (see above).

Mac OS X 10.4 "Tiger" & 10.5 "Leopard"

Cameron Kaiser's TenFourFox is here for you! This browser was forked from a very old version of Firefox, so expect to run into broken websites on occasion. For casual browsing on a secondary computer, however, TenFourFox is perfectly sufficient.

PPC users should download builds from the official website: http://www.floodgap.com/software/tenfourfox/

Intel builds are also available at https://sourceforge.net/projects/tenfourfox/files/unstable/contrib/. These are considered "unofficial", and the author requests that Intel users do not report bugs or ask for support.

Mac OS X 10.6 "Snow Leopard"

InterWeb

With support for WebRTC and an engine based on Firefox 52, Wicknix's InterWeb browser is likely to offer the best compatibility with modern websites on Snow Leopard. Make sure to download the 10.6-specific version: https://randommacstuff.blogspot.com/p/interweb-browser.html

Arctic Fox

If InterWeb runs too slowly on your old hardware, consider trying this other browser from the same developer. It's based on an older engine which won't work with as many websites, but it should be a bit more performant. https://github.com/wicknix/Arctic-Fox/wiki

Mac OS X 10.7 "Lion" & 10.8 "Mountain Lion"

This is the real deal! Blueboxd maintains Chromium Legacy, an up-to-date copy of Chromium (the open source variant of Google Chrome) modified to be compatible with Lion and Mountain Lion. It will work with every website Google Chrome does! https://github.com/blueboxd/chromium-legacy/releases

I have created a "Chromium Legacy Downloader" Preference Pane that makes it a bit easier to install updates: https://jonathanalland.com/downloads/chromium-legacy-downloader.zip

OS X 10.9 "Mavericks"

My favorite version of OS X, and personal daily driver! The best option for Mavericks is Firefox 78 ESR, which is still officially supported by Mozilla with automatic security updates! https://support.mozilla.org/en-US/kb/switch-to-firefox-extended-support-release-esr

Alternately, if you would prefer a Chromium-based browser, Chromium Legacy (see above) is also now compatible with Mavericks.

The short answer is to install FreeBSD or a Linix distro that supports your architecture.

I've done this on a G3 iMac, a G4 XServe and now a (white) 2007 Core 2 Duo MacBook. In each instance, I was able to compile XOrg and get a desktop environment running as well as the latest Chromium and Firefox browsers.

I'm partial to FreeBSD as there is both Intel and PowerPC variants (32 and 64 but versions for each) with a high degree of parity, but Linux is supported as well.

That same Core 2 Duo MacBook has a 10.7.5 partition with WaterFox browser installed and it crashes relentlessly. I've found that older Macs aren't very reliable in running modern browsers. The retro software however, is a different story (anyone remember Karateka?) That works well on older Machines so if you can live without the browser, older versions of macOS/OS X are fine. But if you need browser support - get a newer OS.