If the wires break right at the point where the insulation is stripped, then there's a very good chance that your stripping tool is nicking the wire. All it takes is a tiny scratch on the wire, and it will break at the scratched spot from vibration.
Either adjust your tool so that it doesn't nick the wire, or get a better tool.
Alternatively, use a thermal stripping tool. These use heat to melt the insulation off of the wire. They are available as handtools.
You might also use a tinning pot to remove insulation from wires.
Another thing you need to keep in mind is that stainless steel wire is stiff and springy.
If you have to twist and hold the wire in place against its natural springiness, then it will over time pull itself out of the solder.
You need to arrange your wires so that they are not under mechanical tension or so that the mechanical tension forces the wire into contact with the solder pad .
I've had steel wires pull out. The solution in that case was a tie wrap to hold the whole cable assembly in place such that the one steel wire lay exactly in place when the whole assembly (housing, PCB, and cable) was put together.
Gluing the wires in place will help both with wires breaking and wires pulling out.
To be honest, it sounds very strange that you are using stainless steel wire. I don't know where I could order stainless steel wire.
For connecting breakout boards in a cramped space, I'd probably use 30 AWG Kynar wire wrapping wire.
It is readily available and easy to solder. It isn't as stiff as steel wire, so it won't pull itself out of the soldered joint.
The use of "breakout boards" makes it sound like you may need to reconsider what you are doing. It might be more productive (and more reliable) to have a printed circuit board (PCB) made that can be properly installed rather than wiring in some standard breakout board with jumper wires.
At one place I used to work, we modified electronic equipment by installing small modules inside the equipment.
The modules had connectors with 24AWG wires, and we used them that way if there was room in the equipment.
If there wasn't room, we'd either have a flat flex connector made, or else use 30 AWG Kynar wire wrapping wire to connect them. Many times, the choice fell to wrapping wire because flexes were expensive to have made and difficult to get right.
There were two of use who did the installation work.
The other fellow tried every mechanical wire stripper known to mankind.
I used a melted blob of solder on the tip of my soldering iron to melt the insulation off of the last millimeter of the wire.
His were beautiful, a joy to look at.
Mine were melted, burned, and ugly.
I worked there for over 10 years. We did a lot of that kind of stuff.
The only devices that came back because of broken wires were the beautiful, mechanically stripped wires from my coworker.
The ugly, melted and burned wires that I produced never, ever came back for a broken off wire.
I used the melting technique because way back in the beginning I had tried using the (admittedly crappy) wire strippers we had and found that nicked wires would sometimes break while I was installing the modules. I switched to melting the insulation and never looked back.
The other fellow thought the melted insulation was ugly and unprofessional and kept looking for the perfect setting on the perfect mechanical wire stripper - and kept producing joints that failed in use.