You can slap a coat of paint on just about anything. The question is: Will it last? Unfortunately the answer is no. Considering the amount of water, humidity, and scrubbing an average tub gets, it just won't last.
There are some DIY kits you can find at the apron stores to refinish a tub. It is difficult to get a professional looking finish with them. They should last longer than a coat of paint, but I would still consider them a very temporary solution.
Next is a professional "re-finishing" or "re-glazing". Better than a DIY kit, but still a delicate surface. How long it would last will depend on who does it, your daily use, and how careful you are cleaning it (No abrasive cleansers).
Lastly is a "Re-bath" type company. Haven't done much research on them, but my understanding is they will put a fiberglass or acrylic liner in your existing tub. Probably the most durable solution short of replacing the tub.
Why is that not an option? I would certainly consider replacing an ugly tub if I were redoing the bathroom anyway. It would be cheaper than a rebath. or professional refinish. And a DIY refinish would fail.