Car bearing might work for pipe but it won't work all that well when you're turning a precision piece.
A live center typically uses multiple single row angular contact bearings, preloaded properly to run smoothly under heavy loads. The bearing fits in the housing and arbor are precise in order to maintain proper internal bearing clearances so if you make one, I hope you can bore to really tight tolerances. The tips are usually ground on the housing of the live center in order to maintain concentricity, although they can be turned with fair accuracy.
I would suggest you contact a bearing supplier and ask for their advice on which bearings to use, spacer dimensions that will go between the bearings and then sort out how you're going to preload them. I would also ask for boring tolerances; typically a line-to-line fit will be somewhere in the 3-5 Micron range. Royal uses a needle bearing on the rear of the internal part of the arbor to handle cantilevered loads; not sure if you want to incorporate that.
Most of the difficulty in a project like this is to sort out the bearing choices. And then of course comes the boring and turning. While most of us could do this sort of thing, it isn't quite as simple as throwing any old bearing in there and expecting it to hold tenths tolerances. Sorry, just the truth.
I cannot post jpegs for some reason but I made one for my Sherline lathe. Fun project.