This is one solution for a flush mounted bearing.
You can use a 5 mm wide ball bearing on one leg of it. A ball bearing is preferred here since it more suited for minor thrust loading.
To prevent the bearing from working its way out, it can be cryogenically chilled in liquid nitrogen, and then pressed into place. When the bearing warms up to room temperature it would expand tightly against the metal flat bar causing it to be locked in place.
The same can also be done with the shaft that rides in the bearing, affix the shaft solidly to one flat bar by cryogenically chilling the shaft and pressing into the flat bar. Then press the entire assembly into the inner race of the ball bearing.
Heating the steel flat bar instead of chilling the shaft or bearing can also work for press fitting, as in this case you are expanding the hole beforehand instead of shrinking the part. But to secure the the shaft to the inner race of the bearing by heating, you need to heat the bearing itself, but overheating the bearing will cause it to lose its temper.
If you want to secure the assembly even more you can peen around the outside of the bearing and on the ends of the shaft.