Recommend rec.woodworking for actual answers, but in the interim- (and IMHO, of course), if this is a nice or heirloom chair, I'd take it to a furniture repair place for an estimate, if your town has one. If I wasn't an expert woodworker (and I'm not), I'd only try DIY on a chair that I could live without, or live with it turning out ugly. Busted-off dowel stubs usually have to be drilled out. The dowels usually aren't separate, they are usually the actual ends of the legs, spindles, cross pieces, etc. Only usual exception is if all those items are square-section. Second problem- the holes in seat board, legs, etc, are often not 90-degree angles. Almost impossible to free-hand drill out successfully, leaving a nice tight hole for the new dowel. If the dowel is actually the end of the leg or spindle, grafting a new dowel in there is often impossible. Sometimes it can be faked by gluing the broken ends, and adding a countersunk and puttied screw through the center, from the far end. Often, you have to actually recreate the broken part. This is artisan-level work to make it come out strong and pretty, and expect to pay accordingly. If you can match the chair style at a new or used furniture place, that will probably be cheaper.
So, unless you already have a tilt-table drill press, strap clamps, mortising bits, etc, and a lot of patience, the odds of a successful DIY repair are less than encouraging. As to glue, I'd recommend Gorilla Glue.
What kind of chair? What joint - seat members, leg to seat, etc? Is the dowel part of one member, or a third piece of wood? What diameter is the dowel?
One exception to the above - if the dowel is not blind, that is, if the end of the dowel shows through the piece that it is inserted into when the chair is assembled, you have a pretty good shot at making it work. Just cut it off flush, and knock out the broken stub, and you'll be good to go.
Second possibility - the dowels *are* separate and not part of one of the pieces being joined. I'd suggest getting a slide hammer and driving a small wood screw into the broken dowel(s) and trying to extract them that way, possibly requiring some hand work with a drill bit to clean up the holes afterwards.
Otherwise, I concur with the above summary. It's possible to DIY but (and I really hate when people do this to me, but it's true) if you are asking for advice, you're probably better off taking it somewhere.
If what you are calling "dowels" are the cylindrical sections at the ends of legs or crosspieces and they are at least 5/8" in diameter you might consider doing this:
Drill centered 5/16" diameter holes in the mating broken ends and splint them together by epoxying suitable lengths of 5/16x18 threaded steel rod into those holes. Put some epoxy on the broken wood ends too.
Draw the broken ends tightly together with clamps or a tourniquet while the epoxy cures.
Beware that the three foot dowels sold at HD and Lowes are cut on a shaper with a half-round cutter and will, without exception, be undersized and elliptical in section and will not glue reliably. Dowels made 20, 30 years ago would have been cut with a rotary mill similar in function to a pencil sharpener, and would have been dead round and within a few thou of specified diameter. The bagged, 2" dowels with the spiral glue channels will be okay.
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