Can you achieve a smooth finish when staining press board?

I want to re-stain a small table, but I did not realize it is actually not real wood until I stripped the top. I prepped (sanded, wood conditioner) and stained, but the stain seems to soak into the press board. Any suggestions on achieving a smooth polished finished with stain?

Unless you put some kind of color added hard coating on the pressed board, you can't, simply due to its nature. It probably had the manmade material on it before you started working on it. That is the stuff that starts to peel off, and appears to be a vinyl like material that just happens to have a wood grain texture to it, used in all sorts of pressed board products today.
It would be difficult to reproduce the manufacturer's process and more difficult to even find material to use.
I would consider a natural wood veneer. Bought at Lowe's, Menards, Home Depot, other places, usually comes in various size rolls. You use a laminate glue, or even a wood glue, rolled onto the table top. The veneer is then rolled onto that and pressed down well with a flat weight until it bonds. Then use a dremel tool to trim it to size. Light sanding of the edges--be careful to sand downward, not up and down or the veneer will split at places. Light sanding of the veneer itself. Then new stain to match rest of table and some kind of urethane clear coat over the top. Remember to look carefully, veneers come in oak, luan, and other woods. Each will take stain differently. The harder the wood, the closer the grain will be and the better product you will have when finished.