Well, carbon nanotubes are obviously the stuff of the future, but I don't really know much about those, so I'll be leaving that up to @IsaacKotlicky.
Instead, I'm going to go down a different route, and see if we could integrate steel into your creatures' bones. Now, I'll be talking about endoskeletal bones (ie. the ones on the inside that primarily support your posture and locomotion); there are quite different considerations on exoskeletal bones (eg. turtle shell) and similar not-quite-bone structures (teeth), which I'll touch on at the end.
The word "steel" actually refers to a rather wide variety of (principally) carbon-iron alloys, with an equally wide variety of properties. I wasn't able to find figures for shear stress of steel (presumably because it depends heavily on the geometry), but the yield point (eg. the stress beyond which the material permanently bends, or breaks if it's hard/brittle) seems to be about 10x higher compared to bone for a decent bar of steel you can make into something in your forge.
This is a good thing, because flexibility (expressed as a high yield point) will be a principal requirement for an endoskeletal bone - you want it to be able to take as much stress as possible while flexing, so that (in trauma situations) it absorbs as much of the schock as possible without getting irreversibly bent or broken.
Note that this is the kind of flexibility that we generally talk about in structural engineering - a good bone will be flexible much like a good sword, rather than limp like rubber.
Organic manufacture of steel
Obviously, we won't be able to melt down iron (or iron ore) inside your typical organism. The good news is, we might not have to.
The two chemical processes you need to master is first getting the iron in the first place, and then creating steel and making it into something useful.
Your body can already get iron from food - it's what your red blood cells are made of, among other things - but for volume production, you might want to look for other sources. Iron is one of the most abundant elements, mostly in form of various oxides, so you could concievably have your creatures eat something that contains hematite powder. Extracting atomic iron from hematite is a question of reduction - in industrial practice this is done in a furnace using coke as the reducing agent, but I'm pretty sure any number of organically-available reagents would do the trick.
So now that you have iron, what do you do next? Well, iron (and all other metals, really), have the interesting property that if you simply put together two pieces of it in a favourable chemical environment (such as vacuum, but anything that will prevent the surface from oxidizing will do the trick), they will get cold-welded. This way, you could build up small particles of iron into larger structures - interspersing them with some cementite to up the carbon content - until you get steel.
Altering the properties
The properties of steel are largely dependent on the size and configuration of the monocrystals that make it up (besides the exact chemical composition, of course). In general, larger crystals make harder, more brittle steel, while smaller ones make it softer and more flexible; you want to strike a balance here.
The human body, among others, seems to be capable of growing crystalic substances; that's how it makes enamel out of hydroxylapatite, so I'm going to run with that and assume that we can design an organic process that will grow microcrystals of different sizes.
By depositing and welding them together, you could concievably construct matrices of desireable properties, much like we have inside our bones, but made of steel. If you add cells with the ability to selectively oxidize some parts of that matrix away, you now have osteoblasts and osteoplasts and your steel bones have the same self-repair and adaptation capability as normal bones do.
Exactly what the properties of such a bone would be is anyone's guess, and i depends heavily on the geometry of the bone mesh and the applied forces. We can take the factor of 10 as the ideal case, but other things to consider are that clever geometry can actually make a steel item stronger than its weight/volume would suggest (that's how I-beams work, basically), plus if you can play around with carbon content and crystal sizes, you can mix it up to either mimic pattern welding, or tempering/quenching without the (usually) necessary heat.
I promised to mention something about teeth here, so here goes: where flexibility is needed and you are not expecting to have the object directly scratched or chipped by something really hard, you don't really care about high hardness. Quite the contrary, hardness usually corellates with brittleness, which in turn decreases flexibility, so it's something you might want to avoid.
In some cases however, it might be beneficial. Teeth might be one such case (they are not properly bone, but the enamel - the hard outer part - is also based mainly on hydroxylapatite), where in striking the balance between hardness and brittleness, you might want to go a bit higher on the hardness scale (although not hign anough that you can break your own teeth biting down hard).
Depending on how it's made, steel runs the mohs scale from 4 all the way up to 8. Otherwise, there is plenty of minerals harder than hydroxylapatite that you might grow crystals of using a similar process, although it should be noted that unlike bone, a broken tooth can not be repaired (there are no osteoplasts or osteoblasts that would do that inside a tooth). You could, of course, go the way of the shark and just grow a new one, if you're so inclined.
Finally, note that stronger bones alone do not mean you can put your creature through unlimited abuse - heavy wear or significant shock might damage joints, and you also need good muscle to support the skeleton and help absorb the forces. Also, I hear that bones are supposed to break in order to absorb shock that would cause greater harm to other body parts - making them unbreakable might cause other problems later on.
Well, this was a fun mental excercise; I hope I didn't write anything outrageously wrong:)