You'd get a lot of nutritional adverse effects as other alkali earth metals don't act as exact substitutes for calcium, they have subtly individual properties and would thus behave differently.
Aside from probable problems in growth and bone integrity you could get things like deformities (especially if nutritional levels were very badly skewed during pregnancy) and, more likely, termination of the pregnancy.
Aside from the bones requiring calcium for the hydroxylapatite mineral, they also require other things for the calcification, such as vitamin D and, apparently, small quantities of the other alkali earth metals you listed (although that seems to be debatable). Magnesium, for certain, is an important mineral for humans; not as much is known of strontium and barium.
I am suspecting that while the calcium in bones could be partially substituted by strontium, barium or magnesium due to their superficially similar qualities, the resulting bone mineral would have somewhat different properties than the calcium based apatite - whether it would be harder, more brittle, more flexible, more durable, or just a plain disaster, I cannot say.
What would probably be more important for immediate vitality of the human being are the calcium channels in our nervous system. I don't want to think very much what would happen if there was no calcium available, or if all calcium was replaced by other alkali earth ions.
It should also be noted that it would likely be impossible to offer more of other alkali earth metals than calcium in the diet. Calcium is very ubiquitous, avoiding it would prove immensely problematic, and magnesium, strontium and barium would have to be added in spades...
Actually while you're at it, why not add beryllium and radium to the experiment? I don't think the subject will have a very long or fulfilling life anyway...