Yes it can be a hardware problem.
The reason it doesn't boot off the floppy is probably because the floppy is not set right in the boot order. Or the system is locking up before it gets far enough to boot to it.
Probably what you need to try is unplug any 'extra' hardware from the machine. And PCI cards, sound, your drives etc.. And just have the basics plugged in, which is the power supply, CPU, RAM, and video. Nothing else except power button and front panel connections which are fine.
Blow out any dust in the system. And then you might want to reseat your RAM and video card (if it uses a card). That means just unplugging and replugging back in, to make sure it's making good contact.
If you want to go as far, you can also take your CPU off, clean it and apply new thermal paste, and put it back on making sure it's also tight and good.
If the system still can't work with just those few parts plugged in, one or more of them must be bad. Could still be ports, but more then likely it could be RAM or motherboard, possibly CPU.
The other thing to check, while you're in there, is for any bad capacitors. Those are the little components sticking up from the motherboard. Look like little round barrels. Look at the tops of them and make sure they are all perfectly flat and shiny. If any are bulging even a little, or have redish ooze coming out, then that is likely the culprit. You will have lots of capacitors in there, a bunch by the CPU socket, RAM, and video area, and some more spread out here and there.
Lastly, before throwing in the towel, if you can at least try another power supply, usually they are easy to come by, from another PC, or a friends etc...
Good luck!