Safe inverter circuit to power neon indicators
Two answers:-


The output is floating so it's safer than the mains.  It's like your shaver socket.
You could buy something like a ready made HV psu  It probably has less chance of exploding than a home brew device.  It has both AC and DC outputs.


But another point I'd make is that solderless breadboards are small things for plugging LEDs into.  The gaps between contacts are fairly small, and they won't be rated for 150V+.  Plain vero board might be a better alternative.

I kinda asked something similar when I wanted to mess with some HV stuff here.  There are some useful dimensions there in the answers.