In a perfect world, with perfect electronics, there would be no need for a grounding conductor. It's there to deal with imperfections. The most likely case for a ground fault in this installation would be if the black conductor lost some insulation and shorted itself against the frame of the fan. In such a situation, the casing could become electrified, and a grounded person, touching the fan body could receive a shock.
If it were properly grounded, you'd get a short circuit to the ground wire, and the breaker would trip.
However, the fan is mounted to the ceiling, and the possibility of someone touching it is low. (Unlike, for example, the frame of a fridge or stove, where human contact is part of normal operation.)
So, Is it safe? Quite likely. For added safety, turn the wall switch off when changing bulbs. Is it to code? No. -- Unless, as others have suggested in comments, the mounting box is grounded on the other side.
You can however, locate the breaker, and replace it with a Ground Fault Interrupt breaker, which would trip should a ground fault occur. This isn't something you should do yourself unless you have experience working in the panel.
Another possibility, if your switch happens to be chained off of an outlet, which is quite possible, depending on the routing of the wiring on that circuit, is to replace the outlet with a GFCI outlet, and wire the switch to the load side of it.
Buy a ceiling hugger style fan (the ones without a stem or with the option to mount without the stem).
Mount the fan's bracket directly to the ceiling with toggle bolts or other such fastener. This way the fan is not support by the box at all.
Good luck with your project!
Best Answer
Pictured is a random one dollar 4" octagon box.
That is acceptable for a lamp, but won't do at all for a ceiling fan box.
Since you're rewiring anyway, you'd be better off to spend a latté worth of cost to put a ceiling fan box here and use /3 cable from the switch. That way the next occupant has a choice about putting a ceiling fan here. Obviously the last occupant thought it was worthwhile.