I've run relatively recent Linux distros on hardware with far tighter constraints (like on my Compaq Presario 1210; may you rest in peace, old friend). Damn Small Linux was king here, but distros like Tiny Core and Puppy are also designed for this exact use case.

256MB RAM is actually pretty generous for Windows XP, as is the Pentium III; neither were unheard of in "Made For Windows XP™" systems.