Arch Linux can do that. You need a wired Ethernet NIC that is compatible with iPXE.
Arch is well-suited for this sort of task, since it is specifically aimed at building up a system from the most minimal starting point. It's a lot more work than installing Debian from a USB key or CD, but Debian is aimed at current hardware, which means they're coming from a perspective where even a 512 MB USB key is throwaway technology, whereas from your perspective, that's unattainable dream-level technology. See? Totally incompatible world views.
In principle, you could build up a system based on any Linux distro this way. The trick is that crafting that floppy takes time and expertise. You're not going to find truly generic floppies for ready download, if only because you'd need more space than is available on the floppy for network drivers. So, someone has to cull the set of available Linux network drivers to a reasonable subset, or build the disk specially for the one NIC you happen to be using.
The only reason you don't find Debian netinstall boot floppies on debian.org is that there isn't someone with:
the time
the interest
a floppy drive, and
the technical expertise
Bring all four of those together, and it's a solved problem.
If Arch doesn't work for you, there are a bunch of floppy-based Linux distros. This is not the same as one that boots to a network installer, I know, but once you have a network-connected Linux system up and running, you could build up a Linux system on the HDD from that. It's a stone knives and bearskins sort of task compared to building up a Linux system based on Arch, though.