Electronic – PSU with unspecified max safe amperage

I don't agree with the accepted answer. As to your questions:

1. No it is not completely safe, since it involves high voltage, but it won't burn up unless you short it, and it does provide galvanic isolation so it's less likely to be in a position to deliver a lethal shock.

2. You will get about 11-14V at the point marked "9V". Too low to be reliably regulated down to 9V and a bit high for (linear) regulation down to 5V (but okay for 5V if you don't draw much current).

3. The point marked 170V will actually be at about 240VDC with a light load. That's because the AC voltage is RMS so you get about 1.4 times higher. You're using a 450V cap so it won't hurt the capacitor. You might be better to use an 18V transformer.

4. 1000uF/25V means you if you draw 100mA you'd get about 1V p-p ripple on the 11V line, which is fine for regulation down to 5V.

5. Nixie tubes typically take a few mA. If you have (say) 12mA, the ripple on your 240V line with 45uF will only be a few volts, which is quite acceptable.

I agree that (say) 200K 1/2W bleeder resistor across the 45uF would be a good idea, and a suitable fuse on the input (there is no need to make the fuse more than a few hundred mA).

Take appropriate precautions when dealing with high voltage. Typical precautions include, but are not limited to, not probing the circuit when power is on, using appropriately rated test equipment, keeping everything neat, using a GFI/RCD, discharging the capacitor manually before touching things, even if you think there is a bleeder, working with (at least) one hand in your pocket, etc.

No, don't plug it in. The "bang" was a component getting destroyed. Hopefully a fuse, but likely something else.

The danger (other than getting shocked or starting a fire) is that the PSU's output voltages may no longer be correct. This could destroy your motherboard (bad) or hard drives (worse, in my opinion).

Of course, this may have already happened...

If you have data on your hard drive that you want to recover, I recommend buying a "usb hard drive bay" (or "usb hard drive docking station") (~$15USD) and using another computer to access the data. These bays provide a level of isolation between a possibly-bad hard drive and the computer.

You could simply connect the hard drive directly to a different computer, but this risks damaging the new computer's power supply if the hard drive has developed a short circuit...

After you save your data (or not, if you don't want to), you can simply replace the power supply and see what happens :)