Funnily enough, I believe that's completely achievable with enough time spent on it (probably a few weeks of an engineer working full time).
We have technology like WebVM [1] (from leaningtech / CheerP) or Copy86 [2] that already allows x86 machine code execution/emulation on the web. If you add an OCI client layer on top that is executable in the browser, it should be possible to run Docker containers in the browser.
Sorry I can't reset my docker password to double check, but I think something like this exists [1]. I'm not sure what its limits are, but its a great tool for learning.
I just played with it and it doesn't execute the container in the browser. It executes it remotely and proxies the stdin/stdout and such from/to the browser (the browser is just a client instead of a runner)
I think the biggest challenge here is getting the resources through the browser - e.g. enough memory to run a decent container, or the ability to open a listening port.
