For small beaings and such parts like this your arbor or mandrel can take many forms.
1. Most simple is to put a piece of bar into the chuck and turn to a good firm fit in your bearing and then slip the bearing on with a small amount of loctite or superglue, once machined heat will break the bond. The downsid eto this method with four ends to do is that you have to remove the mandrel each time to heat, it can be marked so it goes back in the same way but not ideal to remove it.
2. Similar to above but turn the spigot that fits into the hole about 2/3rds its length but put a very slight say 1 deg taper on the rest so the diameter increases towards the chuck. You can then slip the bearing onto the parallel part and then wring it onto the taper with a slight twisting action, if light cuts are used it will hold and the part can be wrung off again so mandrel stays put ensuring it's concentric.
3. Again turn a spigot to a good firm fit in the bearing and just shorter than it then drill and tap the end for a bolt or screw to retain the bearing, M5 with a cap head screw would work nicely on these bearings as screw head is just under the diameter you want to turn. Again mandrel does not need to come out of chuck. I'd go with this option.
4. Turn spigot to full length or a bot longer, drill and tap with a taper tap so you get a few full turns of thread. Mark the bit of bar you are using so it can be removed from chuck and then saw down the middle of the spigot before putting back in chuck. You can now slip the bearing onto the spigot and wind in a screw which will expand the spigot as it gets to the tapered part of the thread.
I tend to use aluminium for these sort of things and you soon build up a collection that can be used again and again, maybe skimming down to the next size or shortening.