Cloud solution for an internet cafe? i.e, Minimum hardware.

I want to revamp my internet cafe. I am currently using ten Dell Optiplex GX620 machines as they are very small, compact and somewhat powerful.

Unfortunately, Dell has discontinued this model and the next best model is almost twice as big. This made me wonder if there are any solutions available out there to minimise the hardware used in these setups.

Can you utilise one server(physical or cloud) to serve, say, 10 clients without the clients themselves having any hardware? (Except of course for the monitor, keyboard and mouse) If not, are there any brands out there that do have small desktops? (affordable of course)

Thanks for reading!